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Old 2021-04-05, 01:33   #1068
warachwe
 
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Quote:
For bases which are not prime numbers, this exponent i chosen above is extremely bad in trying to prove the abundance of s(b^i).
Indeed, we know for example that s(15^35) is abundant, but our disproportionately large exponent i does not give an abundant s(15^i) !
It surprises me a lot !
Note also that the more the odd base seems to be the product of a large number of small prime factors (primorial base without the factor 2), the less d seems to have prime factors !
It also surprises me a lot !
The exponent "i" chosen this way is design to make many small primes p divide s(b^i), but this is only work for p dividing neither the base b, nor p divide (q-1) for any of prime q in the factor of b.
So when b is product of many primes, less primes are going to divide s(b^i). (only for "i" construct this way, not for general exponent "i".)

When b is prime, b itself will never divide s(b^i), so "i" chosen as above is efficient.
However, when b is composite, a prime p dividing b still can divide s(b^i) for some i.

Quote:
Let's try to do better for base 3.
If I refer to the explanations of warachwe (https://www.mersenneforum.org/showpo...&postcount=983), I can attempt to build a more efficient i exponent for base 3.
So I built the following exponent j which is much more efficient for the base 3 :

To do this, I took all the prime numbers p of the form 12 * n + 1 and 12 * n-1 for n from 1 to 10,000.
And for each of these prime numbers p, I looked for an odd exponent k such that 3^k == 1 (mod p), for k from 3 to 100,000.
Then, I took as exponent j noted above the lowest common multiple (lcm) of all these odd exponents.
I just realize I make a small mistake on that post.
For p of the form 12 * n + 1, exponent k such that 3^k == 1 (mod p) need not to be odd.
For example, p=37 , no such k exist. That's why 37 is not appear in factor of s(3^i).
For p of the form 12 * n - 1, k = 6 * n -1 work ( so 3^ (6 * n - 1) == 1 (mod p) , but might not be the smallest such k.

Last fiddled with by warachwe on 2021-04-05 at 02:30 Reason: Minor mistake
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Old 2021-04-05, 07:13   #1069
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Thank you warachwe for these new explanations.
Some things seem clearer to me now, and I will be able to further refine my research for exponents.
(I understood of course that it was necessary to replace the "37" of post # 983 by 47. ;-) )
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Old 2021-04-05, 15:14   #1070
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Quote:
Originally Posted by garambois View Post
Let's try to do better for base 3.
If I refer to the explanations of warachwe (https://www.mersenneforum.org/showpo...&postcount=983), I can attempt to build a more efficient i exponent for base 3.
So I built the following exponent j which is much more efficient for the base 3 :
Code:
j = 3^7 * 5^5 * 7^4 * 11^3 * 13^3 * 17^2 * 19^2 * 23^2 * 29^3 * 31^2 * 37^2 * 41^2 * 43^2 * 47^2 * 53^2 * 59 * 61 * 67 * 71^2 * 73^2 * 79 * 83^2 * 89 * 97^2 * 101 * 103 * 107 * 109 * 113 * 127 * 131 * 137 * 139 * 149 * 151 * 157 * 163 * 167 * 173 * 179 * 181 * 191 * 193 * 197 * 199 * 211 * 223 * 227 * 229 * 233 * 239 * 241 * 251 * 257 * 263 * 269 * 271 * 277 * 281 * 283 * 293 * 307 * 311 * 313 * 317 * 331 * 337 * 347 * 349 * 353 * 359 * 367 * 373 * 379 * 383 * 389 * 397 * 401 * 409 * 419 * 421 * 431 * 433 * 439 * 443 * 449 * 457 * 461 * 463 * 467 * 479 * 487 * 491 * 499 * 503 * 509 * 521 * 523 * 541 * 547 * 557 * 563 * 569 * 571 * 577 * 587 * 593 * 599 * 601 * 607 * 613 * 617 * 619 * 631 * 641 * 643 * 647 * 653 * 659 * 661 * 673 * 677 * 683 * 691 * 701 * 709 * 719 * 727 * 733 * 739 * 743 * 751 * 757 * 761 * 769 * 773 * 787 * 797 * 809 * 811 * 821 * 823 * 827 * 829 * 839 * 853 * 857 * 859 * 863 * 877 * 881 * 883 * 887 * 907 * 911 * 919 * 929 * 937 * 941 * 947 * 953 * 967 * 971 * 977 * 983 * 991 * 997 * 1009 * 1013 * 1019 * 1021 * 1031 * 1033 * 1039 * 1049 * 1051 * 1061 * 1063 * 1069 * 1087 * 1091 * 1093 * 1097 * 1103 * 1109 * 1117 * 1123 * 1129 * 1151 * 1153 * 1163 * 1171 * 1181 * 1187 * 1193 * 1201 * 1213 * 1217 * 1223 * 1229 * 1231 * 1237 * 1249 * 1259 * 1277 * 1279 * 1283 * 1289 * 1291 * 1297 * 1301 * 1307 * 1319 * 1321 * 1327 * 1361 * 1367 * 1373 * 1381 * 1399 * 1409 * 1423 * 1427 * 1429 * 1433 * 1439 * 1447 * 1451 * 1453 * 1459 * 1471 * 1481 * 1483 * 1487 * 1489 * 1493 * 1499 * 1511 * 1523 * 1531 * 1543 * 1549 * 1553 * 1559 * 1567 * 1571 * 1579 * 1583 * 1597 * 1601 * 1607 * 1609 * 1613 * 1619 * 1621 * 1627 * 1637 * 1663 * 1667 * 1669 * 1693 * 1697 * 1699 * 1709 * 1721 * 1723 * 1733 * 1741 * 1747 * 1753 * 1759 * 1777 * 1783 * 1789 * 1801 * 1811 * 1823 * 1831 * 1847 * 1861 * 1867 * 1871 * 1873 * 1877 * 1879 * 1889 * 1901 * 1907 * 1913 * 1931 * 1933 * 1951 * 1973 * 1979 * 1987 * 1993 * 1999 * 2003 * 2011 * 2017 * 2027 * 2039 * 2053 * 2063 * 2069 * 2083 * 2087 * 2089 * 2099 * 2113 * 2129 * 2141 * 2143 * 2153 * 2161 * 2179 * 2203 * 2207 * 2213 * 2237 * 2239 * 2251 * 2267 * 2269 * 2273 * 2281 * 2287 * 2297 * 2309 * 2311 * 2333 * 2339 * 2341 * 2351 * 2357 * 2371 * 2377 * 2381 * 2383 * 2389 * 2393 * 2399 * 2411 * 2423 * 2437 * 2441 * 2447 * 2459 * 2467 * 2473 * 2477 * 2503 * 2531 * 2539 * 2543 * 2549 * 2551 * 2557 * 2579 * 2591 * 2593 * 2609 * 2617 * 2621 * 2657 * 2659 * 2663 * 2671 * 2677 * 2683 * 2687 * 2689 * 2693 * 2699 * 2707 * 2711 * 2713 * 2719 * 2729 * 2731 * 2741 * 2753 * 2767 * 2777 * 2789 * 2791 * 2797 * 2801 * 2803 * 2819 * 2833 * 2837 * 2851 * 2857 * 2887 * 2897 * 2903 * 2909 * 2927 * 2939 * 2953 * 2957 * 2963 * 2969 * 2999 * 3001 * 3011 * 3019 * 3023 * 3041 * 3049 * 3061 * 3067 * 3083 * 3109 * 3137 * 3163 * 3167 * 3169 * 3187 * 3203 * 3209 * 3217 * 3221 * 3229 * 3251 * 3253 * 3257 * 3259 * 3271 * 3299 * 3301 * 3307 * 3313 * 3319 * 3323 * 3329 * 3331 * 3343 * 3359 * 3361 * 3389 * 3391 * 3407 * 3413 * 3433 * 3449 * 3457 * 3463 * 3467 * 3491 * 3499 * 3511 * 3517 * 3527 * 3529 * 3533 * 3539 * 3559 * 3571 * 3583 * 3593 * 3613 * 3623 * 3631 * 3643 * 3671 * 3677 * 3691 * 3719 * 3727 * 3761 * 3767 * 3769 * 3779 * 3793 * 3797 * 3803 * 3821 * 3823 * 3851 * 3853 * 3863 * 3881 * 3889 * 3911 * 3919 * 3923 * 3943 * 3947 * 3967 * 3989 * 4003 * 4007 * 4013 * 4019 * 4021 * 4049 * 4051 * 4057 * 4073 * 4079 * 4091 * 4093 * 4099 * 4129 * 4139 * 4153 * 4157 * 4159 * 4177 * 4211 * 4229 * 4231 * 4259 * 4261 * 4271 * 4273 * 4327 * 4337 * 4339 * 4349 * 4363 * 4373 * 4391 * 4397 * 4409 * 4451 * 4457 * 4463 * 4481 * 4483 * 4493 * 4513 * 4517 * 4519 * 4523 * 4547 * 4549 * 4561 * 4583 * 4597 * 4603 * 4621 * 4637 * 4639 * 4643 * 4651 * 4663 * 4673 * 4679 * 4703 * 4733 * 4759 * 4783 * 4787 * 4789 * 4793 * 4799 * 4813 * 4831 * 4861 * 4871 * 4877 * 4889 * 4903 * 4909 * 4919 * 4933 * 4937 * 4943 * 4951 * 4967 * 4973 * 4987 * 4993 * 4999 * 5003 * 5011 * 5023 * 5039 * 5051 * 5059 * 5081 * 5099 * 5101 * 5107 * 5113 * 5171 * 5189 * 5197 * 5209 * 5231 * 5233 * 5273 * 5279 * 5297 * 5303 * 5323 * 5333 * 5393 * 5399 * 5413 * 5419 * 5431 * 5437 * 5441 * 5443 * 5477 * 5501 * 5503 * 5519 * 5563 * 5569 * 5573 * 5623 * 5639 * 5653 * 5659 * 5669 * 5689 * 5711 * 5717 * 5741 * 5779 * 5783 * 5807 * 5821 * 5827 * 5839 * 5849 * 5851 * 5867 * 5879 * 5897 * 5903 * 6029 * 6047 * 6053 * 6089 * 6101 * 6113 * 6121 * 6131 * 6133 * 6151 * 6163 * 6173 * 6199 * 6203 * 6217 * 6221 * 6263 * 6269 * 6287 * 6323 * 6329 * 6361 * 6367 * 6427 * 6449 * 6481 * 6491 * 6521 * 6551 * 6563 * 6569 * 6571 * 6581 * 6599 * 6607 * 6619 * 6679 * 6761 * 6763 * 6779 * 6793 * 6803 * 6869 * 6871 * 6899 * 6947 * 6949 * 6971 * 6983 * 6991 * 7019 * 7027 * 7043 * 7057 * 7079 * 7103 * 7109 * 7121 * 7129 * 7151 * 7159 * 7193 * 7211 * 7219 * 7229 * 7243 * 7283 * 7307 * 7309 * 7321 * 7331 * 7333 * 7349 * 7433 * 7451 * 7457 * 7477 * 7489 * 7499 * 7517 * 7523 * 7529 * 7541 * 7559 * 7561 * 7573 * 7583 * 7603 * 7621 * 7643 * 7649 * 7673 * 7681 * 7687 * 7691 * 7699 * 7703 * 7717 * 7727 * 7741 * 7759 * 7793 * 7823 * 7841 * 7853 * 7879 * 7883 * 7901 * 7937 * 7963 * 7993 * 8011 * 8069 * 8081 * 8093 * 8111 * 8123 * 8161 * 8167 * 8219 * 8231 * 8237 * 8243 * 8273 * 8297 * 8353 * 8363 * 8369 * 8389 * 8419 * 8443 * 8447 * 8501 * 8513 * 8573 * 8599 * 8629 * 8663 * 8677 * 8693 * 8707 * 8731 * 8741 * 8783 * 8819 * 8821 * 8849 * 8863 * 8923 * 8951 * 8969 * 9001 * 9007 * 9029 * 9043 * 9059 * 9091 * 9133 * 9157 * 9181 * 9203 * 9221 * 9281 * 9283 * 9293 * 9319 * 9337 * 9349 * 9371 * 9391 * 9397 * 9419 * 9431 * 9433 * 9473 * 9479 * 9511 * 9539 * 9547 * 9551 * 9619 * 9629 * 9643 * 9689 * 9733 * 9781 * 9787 * 9791 * 9851 * 9871 * 9949 * 10039 * 10061 * 10091 * 10093 * 10111 * 10141 * 10163 * 10177 * 10253 * 10271 * 10313 * 10331 * 10399 * 10529 * 10589 * 10597 * 10613 * 10639 * 10687 * 10691 * 10709 * 10733 * 10781 * 10799 * 10883 * 10939 * 11071 * 11119 * 11149 * 11161 * 11171 * 11173 * 11257 * 11311 * 11317 * 11321 * 11369 * 11393 * 11437 * 11467 * 11471 * 11519 * 11549 * 11579 * 11593 * 11699 * 11719 * 11743 * 11783 * 11801 * 11813 * 11821 * 11831 * 11909 * 11939 * 11959 * 12011 * 12041 * 12101 * 12119 * 12203 * 12263 * 12329 * 12653 * 12671 * 12791 * 12821 * 12899 * 12923 * 12959 * 13001 * 13049 * 13229 * 13313 * 13451 * 13463 * 13553 * 13619 * 13649 * 13763 * 13883 * 13901 * 13913 * 14009 * 14081 * 14153 * 14159 * 14249 * 14303 * 14321 * 14489 * 14561 * 14621 * 14669 * 14699 * 14741 * 14783 * 14831 * 14879 * 14939 * 15101 * 15161 * 15173 * 15233 * 15269 * 15401 * 15569 * 15629 * 15773 * 15791 * 15803 * 15923 * 16001 * 16091 * 16253 * 16301 * 16421 * 16493 * 16553 * 16673 * 16811 * 16823 * 16883 * 16931 * 17159 * 17183 * 17291 * 17333 * 17351 * 17579 * 17669 * 17681 * 17939 * 17981 * 18041 * 18131 * 18149 * 18191 * 18233 * 18341 * 18443 * 18461 * 18731 * 18773 * 18803 * 18899 * 19163 * 19301 * 19319 * 19373 * 19391 * 19433 * 19553 * 19559 * 19661 * 19709 * 19751 * 19889 * 19913 * 19919 * 19991 * 20063 * 20249 * 20369 * 20393 * 20411 * 20441 * 20693 * 20753 * 20759 * 20771 * 20789 * 20879 * 20921 * 20939 * 20963 * 21011 * 21089 * 21149 * 21179 * 21221 * 21341 * 21383 * 21419 * 21611 * 21701 * 21713 * 21803 * 21893 * 22013 * 22079 * 22133 * 22259 * 22271 * 22343 * 22349 * 22409 * 22433 * 22469 * 22481 * 22541 * 22751 * 22853 * 22943 * 23099 * 23279 * 23321 * 23339 * 23459 * 23561 * 23603 * 23669 * 23753 * 23819 * 23909 * 23981 * 24203 * 24239 * 24281 * 24473 * 24509 * 24551 * 24611 * 24683 * 24749 * 24971 * 25073 * 25229 * 25523 * 25601 * 25643 * 25673 * 25703 * 25799 * 25841 * 25913 * 25919 * 26111 * 26189 * 26459 * 26501 * 26573 * 26633 * 26849 * 26879 * 26891 * 26993 * 27143 * 27281 * 27479 * 27539 * 27551 * 27581 * 27743 * 27773 * 27809 * 27893 * 27983 * 28001 * 28019 * 28403 * 28499 * 28559 * 28571 * 28643 * 28751 * 28793 * 28859 * 28901 * 28949 * 28961 * 29021 * 29033 * 29201 * 29339 * 29363 * 29453 * 29483 * 29531 * 29723 * 29873 * 30269 * 30323 * 30389 * 30449 * 30671 * 30689 * 30773 * 30851 * 30983 * 31019 * 31151 * 31253 * 31319 * 31469 * 31649 * 31721 * 31793 * 31799 * 31859 * 32003 * 32009 * 32141 * 32159 * 32381 * 32531 * 32561 * 32573 * 32633 * 32771 * 32789 * 32843 * 32933 * 33023 * 33053 * 33119 * 33179 * 33191 * 33461 * 33479 * 33521 * 33569 * 33623 * 33713 * 33749 * 33773 * 33809 * 33941 * 34253 * 34283 * 34319 * 34439 * 34631 * 34883 * 34913 * 34949 * 35069 * 35081 * 35099 * 35111 * 35291 * 35573 * 35831 * 35933 * 35993 * 35999 * 36083 * 36191 * 36251 * 36353 * 36383 * 36479 * 36563 * 36629 * 36761 * 36791 * 36821 * 36923 * 36929 * 37013 * 37049 * 37139 * 37181 * 37253 * 37379 * 37619 * 37853 * 37871 * 37991 * 38039 * 38183 * 38189 * 38201 * 38231 * 38303 * 38333 * 38453 * 38459 * 38501 * 38639 * 38669 * 38723 * 38861 * 38873 * 38891 * 38933 * 39089 * 39233 * 39239 * 39419 * 39443 * 39521 * 39551 * 39569 * 39659 * 39779 * 39953 * 39971 * 39983 * 39989 * 40193 * 40283 * 40343 * 40559 * 40763 * 40823 * 40853 * 40949 * 41081 * 41231 * 41243 * 41381 * 41399 * 41603 * 41609 * 41621 * 41669 * 41729 * 41969 * 42023 * 42071 * 42089 * 42131 * 42221 * 42359 * 42473 * 42611 * 42719 * 42743 * 42821 * 42923 * 43013 * 43313 * 43391 * 43541 * 43649 * 43661 * 43691 * 43721 * 43793 * 43943 * 44111 * 44129 * 44189 * 44249 * 44273 * 44501 * 44543 * 44651 * 44699 * 44729 * 44879 * 44909 * 45053 * 45119 * 45131 * 45179 * 45263 * 45329 * 45569 * 45599 * 45641 * 45971 * 46181 * 46199 * 46229 * 46349 * 46523 * 46589 * 46619 * 46643 * 46691 * 46703 * 46751 * 47189 * 47279 * 47363 * 47501 * 47513 * 47543 * 47609 * 47639 * 47741 * 48029 * 48131 * 48221 * 48239 * 48413 * 48479 * 48563 * 48593 * 48731 * 48761 * 49103 * 49193 * 49253 * 49331 * 49433 * 49463 * 49481 * 49499 * 49559 * 49811 * 49853 * 49919 * 50021 * 50051 * 50261 * 50273 * 50411 * 50423 * 50513 * 50591 * 50741 * 50873 * 50969 * 50993 * 51203 * 51503 * 51521 * 51539 * 51659 * 51893 * 52103 * 52121 * 52163 * 52289 * 52361 * 52379 * 52511 * 52553 * 52571 * 52583 * 52631 * 52733 * 52883 * 53051 * 53093 * 53309 * 53411 * 53453 * 53549 * 53591 * 53639 * 53849 * 53951 * 54011 * 54101 * 54251 * 54293 * 54401 * 54413 * 54443 * 54773 * 54941 * 54959 * 55229 * 55439 * 55469 * 55631 * 55661 * 55673 * 55721 * 55733 * 55799 * 55829 * 55889 * 55931 * 56009 * 56081 * 56099 * 56123 * 56393 * 56489 * 56519 * 56531 * 56663 * 56681 * 56783 * 56891 * 56909 * 56921 * 56951 * 57041 * 57149 * 57203 * 57329 * 57413 * 57773 * 57839 * 57881 * 58013 * 58049 * 58193 * 58211 * 58451 * 58511 * 58601 * 58889 * 58979 * 59021 * 59063 * 59123 * 59369 * 59393 * 59399 * 59453 * 59513 * 59621 * 59723 * 59879 * 59981
To do this, I took all the prime numbers p of the form 12 * n + 1 and 12 * n-1 for n from 1 to 10,000.
And for each of these prime numbers p, I looked for an odd exponent k such that 3^k == 1 (mod p), for k from 3 to 100,000.
Then, I took as exponent j noted above the lowest common multiple (lcm) of all these odd exponents.
Finally, I just had to enter this exponent for base 3 in Oliver's program :
mono patf.exe -v 3 "j" 100000
Then I tested the abundance of the d given by the program.
Result : abundance = 0.806
I also tried this entry :
mono patf.exe -v 3 "j" 1000000
Then I tested the abundance of the d given by the program.
Result : abundance = 0.877
I also tried this entry :
mono patf.exe -v 3 "j" 10000000
It is impossible for me to test the abundance of the divisor d given by the program !

But now, for the first time, with this exponent j, I'm starting to believe that maybe we have abundance for base 3.
At least, I hope so, otherwise I'll have to look for an exponent k !!!

Who will dare to enter this instruction below into Oliver's program or Happy's new program to verify it ?
mono patf.exe -v 3 "j" 1000000000
I dared, and I came up short. It took two hours with 8 threads on my Kubuntu Focus, and it generated over 39 MB of factors. More than half of the time was spent on the sigma calculation Thanks to changes to my code (available on GitHub) that also output the abundance ratio (I also added changes to the argument passing and the ability to pass the exponent via a file), I was able to grep the output (which had been redirected to a file) and see this:

Code:
Index 1 of 3^5722[...]9375 is not abundant. (0.96466)
Sorry. It only improved from 0.936224 at 30000000, so I think you'll have to go for k.

Last fiddled with by Happy5214 on 2021-04-05 at 15:16
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Old 2021-04-05, 17:28   #1071
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Quote:
Originally Posted by Happy5214 View Post
I dared, and I came up short. It took two hours with 8 threads on my Kubuntu Focus, and it generated over 39 MB of factors. More than half of the time was spent on the sigma calculation Thanks to changes to my code (available on GitHub) that also output the abundance ratio (I also added changes to the argument passing and the ability to pass the exponent via a file), I was able to grep the output (which had been redirected to a file) and see this:

Code:
Index 1 of 3^5722[...]9375 is not abundant. (0.96466)
Sorry. It only improved from 0.936224 at 30000000, so I think you'll have to go for k.

39 MB of factors !!!
Sorry, I didn't know the amount would be so impressive !
I don't know if we're still in the realm of reason, but on the other hand, we don't seem very far off the mark.
It is also possible that we will have to wait for the next generation of computers to complete this research !
I will try to generate an even more efficient exponent k, but it will also be larger and therefore, it will be even more difficult to process.
It will take me a few days all the same ...
And I don't know if we will gain much on the abundance with this new exponent ?
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Old 2021-04-05, 19:24   #1072
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Quote:
Originally Posted by garambois View Post
39 MB of factors !!!
Sorry, I didn't know the amount would be so impressive !
I don't know if we're still in the realm of reason, but on the other hand, we don't seem very far off the mark.
It is also possible that we will have to wait for the next generation of computers to complete this research !
I will try to generate an even more efficient exponent k, but it will also be larger and therefore, it will be even more difficult to process.
It will take me a few days all the same ...
And I don't know if we will gain much on the abundance with this new exponent ?
If it interests anyone, the abundance ratio at 1e7 was 0.921549, and the difference between the ratios at 1e6 and 1e7 (~0.04455) is very close to the difference between the ratios at 1e7 and 1e9 (.043111). I don't know if we can use that to project ahead, or if this is simply coincidence.
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Old 2021-04-06, 07:23   #1073
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Quote:
Originally Posted by Happy5214 View Post
If it interests anyone, the abundance ratio at 1e7 was 0.921549, and the difference between the ratios at 1e6 and 1e7 (~0.04455) is very close to the difference between the ratios at 1e7 and 1e9 (.043111). I don't know if we can use that to project ahead, or if this is simply coincidence.

If we try to guess :
1e5 : abundance = 0.806
1e6 : abundance = 0.877
1e7 : abundance = 0.921
1e9 : abundance = 0.964
1e13 : abundance =1

No one will be able to test the exponent j for d with prime numbers up to 1e13.
I am building a new exponent hoping to improve the abundance enough to get to 1 for prime numbers up to 1e9.
But the construction will take at least 60 hours.
I started Monday April 5 in the evening at 9:00 p.m.
So I won't have this exponent until late Thursday afternoon, or Friday April 9th.
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Old 2021-04-07, 10:20   #1074
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One new conjecture


This new conjecture completes and replaces conjectures (134) and (135).
k is an integer.


Conjecture (140) :
If a base b = p# is primorial (p prime > 41), then s(b^(2 + 6 * k)) is abundant.


Let me present you the data that allowed me to state this conjecture, then give you some explanations and finally ask you a curious question following a curious observation, in bold at the bottom of the post (paragraph "For b = 3# to b = 5#").


Data (Tables not complete presented here, because too long) :

Code:
b = 3# = 6 = 2 * 3
[720, 1080, 1260, 1440, 1680, 1800, 1980, 2160, 2340, 2520, 2640, 2700, 2772, 2880, 3120, 3240, 3360, 3600, 3780, 3960, 4032, 4200, 4320, 4620, 4680, 4752, 5040, 5280, 5400, 5460, 5760, 5880, 5940, 6120, 6240, 6300, 6480, 6552, 6720, 6840, 7020, 7200, 7392, 7560, 7920, 8100, 8280, 8400, 8640, 8820, 9000, 9072, 9240, 9360, 9720, 9900, 10080, 10440, 10560, 10800, 10920, 11340, 11520, 11760, 11880, 12240, 12480, 12600, 12960, 13104, 13440, 13680, 13860, 14040, 14400, 15120, 15840, 16200, 16380, 16560, 16800, 17280, 17640, 17820, 18000, 18360, 18480, 18720, 18900, 19440, 19800, 20160, 20520, 20880, 21060, 21120, 21600, 21840, 22680, 23760, 24300, 24840, 25200, 25920, 26460, 27000, 27720, 28080, 29160, 29700, 30240, 31320, 31680, 32400, 32760]

b = 5# = 30 = 2 * 3 * 5
[55440, 65520, 75600, 80640, 90720, 100800, 110880, 120960, 131040, 138600, 151200, 166320, 171360, 181440, 196560, 201600, 221760, 226800, 241920, 262080, 272160, 277200, 302400, 332640, 342720, 362880, 393120, 403200, 415800, 453600, 498960, 514080, 544320, 589680, 604800]

b = 7# = 210 = 2 * 3 * 5 * 7
[80, 440, 1520, 1700, 1808, 2240, 2960, 3248, 3680, 3968, 4400, 6560, 6848, 7280, 7568, 8000, 8288, 8360, 8720, 8828, 8960, 9800, 10340, 10880, 11600, 12320, 12608, 13040, 13328, 13580, 14300, 14768, 14960, 15200, 15488, 16280, 16640, 16928, 17360, 18080, 18368, 18656, 19520, 20240, 21680, 22400, 22688, 23120, 23408, 23840, 24128, 24200, 24920, 25460, 26000, 26720, 27440, 27728, 28160, 29240, 29348, 29600, 29888, 30320, 30800, 31328, 31760, 32480, 33200, 33440, 33488, 34100, 34640, 35288, 35360, 36080, 36800, 37520, 38528, 38960, 39248, 39320, 40040, 41120, 41840, 42560, 42848, 43010, 43280, 44000, 44720, 45008, 45080, 45440, 45920, 46880, 47168, 47600, 47888, 48608, 49280, 49328, 49760, 51128, 51200, 51920, 52640, 53360, 53648, 53900, 54080, 54368, 55088, 55160, 55880, 56240, 56672, 56960, 57200, 57680, 57968, 58400, 58688, 59120, 59840, 60560, 61100, 62000, 62288, 62720, 63008, 63440, 63800, 64400, 64448, 64880, 64988, 65600, 65780, 66608, 67040, 67760, 68480, 68768, 69020, 69200, 69920, 70400, 70928, 71360, 72080, 72800, 73088, 73520, 73808, 74240, 74600, 74960, 75680, 76544, 76868, 77120, 77840, 78128, 78560, 78848, 79640, 80000, 80108, 80900, 81620, 82160, 82880, 83168, 83600, 83888, 84320, 85400, 86048, 86240, 86480, 86768, 87200, 87560, 87920, 88208, 88640, 89180, 90728, 90800, 91520, 91988, 92240, 92960, 93500, 93968, 94688, 94760, 95120, 95480, 96200, 96740, 97280, 98000, 98720, 99008, 99440]

b = 11# = 2310 = 2 * 3 * 5 * 7 * 11
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b = 13# = 30030 = 2 * 3 * 5 * 7 * 11 * 13
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b = 17# = 510510 = 2 * 3 * 5 * 7 * 11 * 13 * 17
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b = 19# = 9699690 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19
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b = 23# = 223092870 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23
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b = 29# = 6469693230 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29
[14, 20, 26, 32, 44, 56, 62, 68, 74, 80, 92, 98, 104, 110, 116, 122, 134, 140, 146, 152, 164, 176, 188, 194, 200, 212, 224, 230, 242, 248, 254, 260, 272, 284, 290, 296, 314, 320, 332, 344, 350, 356, 362, 368, 374, 386, 392, 398, 404, 416, 428, 434, 440, 464, 476, 482, 494, 500, 506, 512, 524, 530, 536, 554, 560, 566, 572, 584, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 674, 680, 692, 704, 710, 716, 722, 728, 734, 752, 758, 764, 770, 776, 782, 788, 794, 800, 812, 818, 824, 830, 836, 842, 848, 854, 860, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 956, 962, 968, 974, 980, 986, 1004, 1016, 1022, 1028, 1034, 1040, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1112, 1124, 1148, 1154, 1160, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1232, 1238, 1244, 1250, 1256, 1274, 1280, 1286, 1292, 1304, 1322, 1328, 1334, 1340, 1352, 1358, 1364, 1376, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1472, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1532, 1538, 1544, 1550, 1562, 1568, 1574, 1580, 1592, 1604, 1610, 1616, 1628, 1634, 1640, 1652, 1658, 1664, 1682, 1688, 1694, 1700, 1724, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1844, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1904, 1910, 1922, 1934, 1940, 1946, 1952, 1964, 1976, 1988, 1994, 2000, 2012, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2072, 2078, 2084, 2096, 2114, 2120, 2132, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2192, 2204, 2210, 2228, 2234, 2240, 2246, 2258, 2264, 2276, 2282, 2288, 2294, 2300, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2372, 2378, 2384, 2402, 2408, 2414, 2420, 2438, 2444, 2450, 2456, 2474, 2480, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2552, 2564, 2576, 2582, 2588, 2594, 2600, 2606, 2618, 2624, 2636, 2642, 2654, 2660, 2666, 2672, 2684, 2690, 2696, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2792, 2804, 2816, 2834, 2840, 2852, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2912, 2918, 2924, 2936, 2948, 2954, 2960, 2978, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3032, 3044, 3050, 3056, 3068, 3074, 3080, 3086, 3092, 3104, 3116, 3122, 3128, 3134, 3140, 3146, 3158, 3164, 3170, 3194, 3200, 3212, 3224, 3230, 3236, 3242, 3248, 3254, 3266, 3272, 3284, 3290, 3296, 3308, 3314, 3320, 3332, 3338, 3344, 3356, 3362, 3374, 3380, 3392, 3398, 3404, 3410, 3416, 3422, 3434, 3440, 3452, 3464, 3482, 3488, 3494, 3500, 3506, 3512, 3524, 3530, 3536, 3542, 3548, 3554, 3560, 3584, 3590, 3596, 3602, 3608, 3614, 3632, 3644, 3662, 3668, 3674, 3680, 3692, 3698, 3704, 3710, 3716, 3722, 3728, 3734, 3740, 3752, 3758, 3764, 3770, 3776, 3794, 3800, 3806, 3812, 3818, 3824, 3836, 3842, 3848, 3854, 3860, 3872, 3884, 3896, 3914, 3920, 3926, 3944, 3956, 3962, 3968, 3974, 3992, 4004, 4010, 4022, 4028, 4034, 4040, 4052, 4058, 4064, 4076, 4082, 4088, 4094, 4100, 4112, 4118, 4124, 4130, 4136, 4148, 4154, 4160, 4172, 4178, 4184, 4190, 4202, 4208, 4214, 4220, 4232, 4238, 4244, 4250, 4256, 4262, 4274, 4280, 4292, 4304, 4310, 4316, 4322, 4334, 4340, 4346, 4352, 4364, 4370, 4376, 4388, 4394, 4400, 4424, 4430, 4436, 4442, 4454, 4460, 4466, 4472, 4484, 4496, 4508, 4514, 4520, 4526, 4532, 4544, 4550, 4556, 4562, 4568, 4574, 4580, 4586, 4592, 4598, 4604, 4616, 4628, 4634, 4640, 4646, 4652, 4664, 4676, 4682, 4688, 4694, 4712, 4718, 4724, 4748, 4754, 4760, 4766, 4778, 4784, 4796, 4802, 4808, 4814, 4820, 4832, 4844, 4850, 4856, 4874, 4880, 4886, 4892, 4904, 4916, 4922, 4928, 4934, 4940, 4964, 4970, 4976, 4982, 4988, 4994, 5000]

b = 31# = 200560490130 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31
[2, 8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 668, 674, 680, 686, 692, 698, 704, 710, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1100, 1106, 1112, 1118, 1124, 1130, 1136, 1142, 1148, 1154, 1160, 1166, 1172, 1178, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1226, 1232, 1238, 1244, 1250, 1256, 1262, 1268, 1274, 1280, 1286, 1292, 1298, 1304, 1310, 1316, 1322, 1328, 1334, 1340, 1346, 1352, 1358, 1364, 1370, 1376, 1382, 1388, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1460, 1466, 1472, 1478, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1526, 1532, 1538, 1544, 1550, 1556, 1562, 1568, 1574, 1580, 1586, 1592, 1598, 1604, 1610, 1616, 1622, 1628, 1634, 1640, 1646, 1652, 1658, 1664, 1670, 1676, 1682, 1688, 1694, 1700, 1706, 1712, 1718, 1724, 1730, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1778, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1838, 1844, 1850, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1898, 1904, 1910, 1916, 1922, 1928, 1934, 1940, 1946, 1952, 1958, 1964, 1970, 1976, 1982, 1988, 1994, 2000, 2006, 2012, 2018, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2066, 2072, 2078, 2084, 2090, 2096, 2102, 2108, 2114, 2120, 2126, 2132, 2138, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2186, 2192, 2198, 2204, 2210, 2216, 2222, 2228, 2234, 2240, 2246, 2252, 2258, 2264, 2270, 2276, 2282, 2288, 2294, 2300, 2306, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2366, 2372, 2378, 2384, 2390, 2396, 2402, 2408, 2414, 2420, 2426, 2432, 2438, 2444, 2450, 2456, 2462, 2468, 2474, 2480, 2486, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2546, 2552, 2558, 2564, 2570, 2576, 2582, 2588, 2594, 2600, 2606, 2612, 2618, 2624, 2630, 2636, 2642, 2648, 2654, 2660, 2666, 2672, 2678, 2684, 2690, 2696, 2702, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2786, 2792, 2798, 2804, 2810, 2816, 2822, 2828, 2834, 2840, 2846, 2852, 2858, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2906, 2912, 2918, 2924, 2930, 2936, 2942, 2948, 2954, 2960, 2966, 2972, 2978, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3026, 3032, 3038, 3044, 3050, 3056, 3062, 3068, 3074, 3080, 3086, 3092, 3098, 3104, 3110, 3116, 3122, 3128, 3134, 3140, 3146, 3152, 3158, 3164, 3170, 3176, 3182, 3188, 3194, 3200, 3206, 3212, 3218, 3224, 3230, 3236, 3242, 3248, 3254, 3260, 3266, 3272, 3278, 3284, 3290, 3296, 3302, 3308, 3314, 3320, 3326, 3332, 3338, 3344, 3350, 3356, 3362, 3368, 3374, 3380, 3386, 3392, 3398, 3404, 3410, 3416, 3422, 3428, 3434, 3440, 3446, 3452, 3458, 3464, 3470, 3476, 3482, 3488, 3494, 3500, 3506, 3512, 3518, 3524, 3530, 3536, 3542, 3548, 3554, 3560, 3566, 3572, 3578, 3584, 3590, 3596, 3602, 3608, 3614, 3620, 3626, 3632, 3638, 3644, 3650, 3656, 3662, 3668, 3674, 3680, 3686, 3692, 3698, 3704, 3710, 3716, 3722, 3728, 3734, 3740, 3746, 3752, 3758, 3764, 3770, 3776, 3782, 3788, 3794, 3800, 3806, 3812, 3818, 3824, 3830, 3836, 3842, 3848, 3854, 3860, 3866, 3872, 3878, 3884, 3890, 3896, 3902, 3908, 3914, 3920, 3926, 3932, 3938, 3944, 3950, 3956, 3962, 3968, 3974, 3980, 3986, 3992, 3998, 4004, 4010, 4016, 4022, 4028, 4034, 4040, 4046, 4052, 4058, 4064, 4070, 4076, 4082, 4088, 4094, 4100, 4106, 4112, 4118, 4124, 4130, 4136, 4142, 4148, 4154, 4160, 4166, 4172, 4178, 4184, 4190, 4196, 4202, 4208, 4214, 4220, 4226, 4232, 4238, 4244, 4250, 4256, 4262, 4268, 4274, 4280, 4286, 4292, 4298, 4304, 4310, 4316, 4322, 4328, 4334, 4340, 4346, 4352, 4358, 4364, 4370, 4376, 4382, 4388, 4394, 4400, 4406, 4412, 4418, 4424, 4430, 4436, 4442, 4448, 4454, 4460, 4466, 4472, 4478, 4484, 4490, 4496, 4502, 4508, 4514, 4520, 4526, 4532, 4538, 4544, 4550, 4556, 4562, 4568, 4574, 4580, 4586, 4592, 4598, 4604, 4610, 4616, 4622, 4628, 4634, 4640, 4646, 4652, 4658, 4664, 4670, 4676, 4682, 4688, 4694, 4700, 4706, 4712, 4718, 4724, 4730, 4736, 4742, 4748, 4754, 4760, 4766, 4772, 4778, 4784, 4790, 4796, 4802, 4808, 4814, 4820, 4826, 4832, 4838, 4844, 4850, 4856, 4862, 4868, 4874, 4880, 4886, 4892, 4898, 4904, 4910, 4916, 4922, 4928, 4934, 4940, 4946, 4952, 4958, 4964, 4970, 4976, 4982, 4988, 4994, 5000]

b = 37# = 7420738134810 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37
[8, 14, 20, 26, 32, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 668, 674, 680, 686, 692, 698, 704, 710, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1082, 1088, 1094, 1100, 1106, 1112, 1118, 1124, 1130, 1136, 1142, 1148, 1154, 1160, 1166, 1172, 1178, 1184, 1190, 1196, 1202, 1208, 1214, 1220, 1226, 1232, 1238, 1244, 1250, 1256, 1262, 1268, 1274, 1280, 1286, 1292, 1298, 1304, 1310, 1316, 1322, 1328, 1334, 1340, 1346, 1352, 1358, 1364, 1370, 1376, 1382, 1388, 1394, 1400, 1406, 1412, 1418, 1424, 1430, 1436, 1442, 1448, 1454, 1460, 1466, 1472, 1478, 1484, 1490, 1496, 1502, 1508, 1514, 1520, 1526, 1532, 1538, 1544, 1550, 1556, 1562, 1568, 1574, 1580, 1586, 1592, 1598, 1604, 1610, 1616, 1622, 1628, 1634, 1640, 1646, 1652, 1658, 1664, 1670, 1676, 1682, 1688, 1694, 1700, 1706, 1712, 1718, 1724, 1730, 1736, 1742, 1748, 1754, 1760, 1766, 1772, 1778, 1784, 1790, 1796, 1802, 1808, 1814, 1820, 1826, 1832, 1838, 1844, 1850, 1856, 1862, 1868, 1874, 1880, 1886, 1892, 1898, 1904, 1910, 1916, 1922, 1928, 1934, 1940, 1946, 1952, 1958, 1964, 1970, 1976, 1982, 1988, 1994, 2000, 2006, 2012, 2018, 2024, 2030, 2036, 2042, 2048, 2054, 2060, 2066, 2072, 2078, 2084, 2090, 2096, 2102, 2108, 2114, 2120, 2126, 2132, 2138, 2144, 2150, 2156, 2162, 2168, 2174, 2180, 2186, 2192, 2198, 2204, 2210, 2216, 2222, 2228, 2234, 2240, 2246, 2252, 2258, 2264, 2270, 2276, 2282, 2288, 2294, 2300, 2306, 2312, 2318, 2324, 2330, 2336, 2342, 2348, 2354, 2360, 2366, 2372, 2378, 2384, 2390, 2396, 2402, 2408, 2414, 2420, 2426, 2432, 2438, 2444, 2450, 2456, 2462, 2468, 2474, 2480, 2486, 2492, 2498, 2504, 2510, 2516, 2522, 2528, 2534, 2540, 2546, 2552, 2558, 2564, 2570, 2576, 2582, 2588, 2594, 2600, 2606, 2612, 2618, 2624, 2630, 2636, 2642, 2648, 2654, 2660, 2666, 2672, 2678, 2684, 2690, 2696, 2702, 2708, 2714, 2720, 2726, 2732, 2738, 2744, 2750, 2756, 2762, 2768, 2774, 2780, 2786, 2792, 2798, 2804, 2810, 2816, 2822, 2828, 2834, 2840, 2846, 2852, 2858, 2864, 2870, 2876, 2882, 2888, 2894, 2900, 2906, 2912, 2918, 2924, 2930, 2936, 2942, 2948, 2954, 2960, 2966, 2972, 2978, 2984, 2990, 2996, 3002, 3008, 3014, 3020, 3026, 3032, 3038, 3044, 3050, 3056, 3062, 3068, 3074, 3080, 3086, 3092, 3098, 3104, 3110, 3116, 3122, 3128, 3134, 3140, 3146, 3152, 3158, 3164, 3170, 3176, 3182, 3188, 3194, 3200, 3206, 3212, 3218, 3224, 3230, 3236, 3242, 3248, 3254, 3260, 3266, 3272, 3278, 3284, 3290, 3296, 3302, 3308, 3314, 3320, 3326, 3332, 3338, 3344, 3350, 3356, 3362, 3368, 3374, 3380, 3386, 3392, 3398, 3404, 3410, 3416, 3422, 3428, 3434, 3440, 3446, 3452, 3458, 3464, 3470, 3476, 3482, 3488, 3494, 3500, 3506, 3512, 3518, 3524, 3530, 3536, 3542, 3548, 3554, 3560, 3566, 3572, 3578, 3584, 3590, 3596, 3602, 3608, 3614, 3620, 3626, 3632, 3638, 3644, 3650, 3656, 3662, 3668, 3674, 3680, 3686, 3692, 3698, 3704, 3710, 3716, 3722, 3728, 3734, 3740, 3746, 3752, 3758, 3764, 3770, 3776, 3782, 3788, 3794, 3800, 3806, 3812, 3818, 3824, 3830, 3836, 3842, 3848, 3854, 3860, 3866, 3872, 3878, 3884, 3890, 3896, 3902, 3908, 3914, 3920, 3926, 3932, 3938, 3944, 3950, 3956, 3962, 3968, 3974, 3980, 3986, 3992, 3998, 4004, 4010, 4016, 4022, 4028, 4034, 4040, 4046, 4052, 4058, 4064, 4070, 4076, 4082, 4088, 4094, 4100, 4106, 4112, 4118, 4124, 4130, 4136, 4142, 4148, 4154, 4160, 4166, 4172, 4178, 4184, 4190, 4196, 4202, 4208, 4214, 4220, 4226, 4232, 4238, 4244, 4250, 4256, 4262, 4268, 4274, 4280, 4286, 4292, 4298, 4304, 4310, 4316, 4322, 4328, 4334, 4340, 4346, 4352, 4358, 4364, 4370, 4376, 4382, 4388, 4394, 4400, 4406, 4412, 4418, 4424, 4430, 4436, 4442, 4448, 4454, 4460, 4466, 4472, 4478, 4484, 4490, 4496, 4502, 4508, 4514, 4520, 4526, 4532, 4538, 4544, 4550, 4556, 4562, 4568, 4574, 4580, 4586, 4592, 4598, 4604, 4610, 4616, 4622, 4628, 4634, 4640, 4646, 4652, 4658, 4664, 4670, 4676, 4682, 4688, 4694, 4700, 4706, 4712, 4718, 4724, 4730, 4736, 4742, 4748, 4754, 4760, 4766, 4772, 4778, 4784, 4790, 4796, 4802, 4808, 4814, 4820, 4826, 4832, 4838, 4844, 4850, 4856, 4862, 4868, 4874, 4880, 4886, 4892, 4898, 4904, 4910, 4916, 4922, 4928, 4934, 4940, 4946, 4952, 4958, 4964, 4970, 4976, 4982, 4988, 4994, 5000]

b = 41# = 304250263527210 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41
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b = 43# = 13082761331670030 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43
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b = 47# = 614889782588491410 = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 47
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...
...

b = 97# = 100# = 2305567963945518424753102147331756070
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...
...

b = 997# = 1000# = 19590340644999083431262508198206381046123972390589368223882605328968666316379870661851951648789482321596229559115436019149189529725215266728292282990852649023362731392404017939142010958261393634959471483757196721672243410067118516227661133135192488848989914892157188308679896875137439519338903968094905549750386407106033836586660683539201011635917900039904495065203299749542985993134669814805318474080581207891125910
[2, 8, 14, 20, 26, 32, 34, 38, 44, 50, 56, 62, 68, 74, 80, 86, 92, 98, 104, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 174, 176, 182, 188, 194, 200, 206, 212, 218, 224, 230, 236, 242, 244, 248, 254, 260, 266, 272, 278, 284, 290, 296, 302, 308, 314, 320, 326, 332, 338, 344, 350, 356, 362, 368, 374, 380, 384, 386, 392, 398, 404, 410, 416, 422, 428, 434, 440, 446, 452, 454, 458, 464, 470, 476, 482, 488, 494, 500, 506, 512, 518, 524, 530, 536, 542, 548, 554, 560, 566, 572, 578, 584, 590, 594, 596, 602, 608, 614, 620, 626, 632, 638, 644, 650, 656, 662, 664, 668, 674, 680, 686, 692, 698, 704, 710, 714, 716, 722, 728, 734, 740, 746, 752, 758, 764, 770, 776, 782, 788, 794, 800, 804, 806, 812, 818, 824, 830, 836, 842, 848, 854, 860, 866, 872, 874, 878, 884, 890, 896, 902, 908, 914, 920, 926, 932, 938, 944, 950, 956, 962, 968, 974, 980, 986, 992, 998, 1004, 1010, 1014, 1016, 1022, 1028, 1034, 1040, 1046, 1052, 1058, 1064, 1070, 1076, 1084, 1328, 1504, 1748, 1924]
In this table, for all primorial bases b = p# for p (prime) from 3 to 47, one can see all exponents e ranked in order, up to a certain limit, such that s(b^e) is abundant.
At the end of the table, we have added two very large primorial bases : b = 97# and b = 997#.


Some explanations, remarks and a curious question (in bold) :

Let's describe what we observe, starting at the bottom of the table.

For b = 97# and b = 997#.
We see that all exponents e = 2 + 6 * k, (k integer) are present.
There are even other exponents that are not of the form 2 + 6 * k that appear in addition.
For example, for b = 97#, we have three extra exponents which are : 454, 2274 and 3184.
For b = 997#, we have a larger list : 34, 174, 244, 384, 454, 594, 664, 714, 804, 874, 1014, 1504, 1924.
But I was totally unable to formulate a conjecture that says what these added numbers are !

For b = 7# to b = 47#.
All the exponents we see in the table are of the form e = 2 + 6 * k, without exception.
But I think this would not be true if we were considering very large exponents with very large k. Exponents of some other form would most likely be added.
On the other hand, we can note that only from b = 43# onwards all exponents of the form 2 + 6 * k appear in the table, hence the formulation of the conjecture (140) which specifies "(p prime >41)".
Let us see for which bases some exponents of the form 2 + 6 * k are missing and for which we see them all :
For b = 47#, all exponents are present.
For b = 43#, all exponents are present.
For b = 41#, the exponent e = 2 is missing (k = 0)
For b = 37#, the exponent e = 2 is missing (k = 0)
For b = 31#, all exponents are present.
For b = 29#, many exponents are missing : e = 2 (k = 0), e = 8 (k = 1), e = 38 (k = 6), e = 50 (k = 8), ...
For b = 23#, even more exponents are missing : e = 2 (k = 0), e = 8 (k = 1), e = 20 (k = 3), ...
And there are always more exponents of the form 2 + 6 * k missing for b = p# when p decreases to 7.
...
...
For b = 7#, almost all exponents of the form e = 2 + 6 * k are missing and there, it is the opposite, there are only very few present : e = 80 (k = 13), e = 440 (k = 73), e = 1520 (k = 253), e = 1700 (k = 283), ...

For b = 3# to b = 5#.
The primorial numbers 6 = 3# and 30 = 5# really behave very differently from the other primorial numbers !!!
It's just the opposite. None of the exponents present in the table are of the form e = 2 + 6 * k and this is very curious !
Why does everything switch when we go from the base b = 5# to the base 7# ?
I did not succeed in stating a conjecture which allows to predict what are the exponents e present in the table for these two very special bases !
For the base b = 3#, there seem to be exponents of the form e = 720 * k and e = 1080 * k, k integer, but, I did not check this.

For b = 2#.
Refer to the conjecture (137).
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Old 2021-04-08, 17:42   #1075
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Page updated.
Many thanks to all for your help !

I extended almost thirty bases, because yoyo had done a lot of calculations that did not appear on the page.
I also extended some bases that were not reserved by yoyo to harmonize the number of exponents for each base.
For example, now, for all the bases from 20 to 100, we can see all the exponents up to 100 appear in the table.
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Old 2021-04-10, 19:46   #1076
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Another merge to report:

Code:
46^65:i1037 merges with 2710248:i240
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Old 2021-04-10, 22:15   #1077
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And another termination. 220^55 terminates at 59.
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Old 2021-04-11, 15:45   #1078
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Quote:
Originally Posted by Happy5214 View Post
Another merge to report:
Code:
46^65:i1037 merges with 2710248:i240
Quote:
Originally Posted by Happy5214 View Post
And another termination. 220^55 terminates at 59.
OK, Many thanks.
This information will be taken into account in the next update.


Quote:
Originally Posted by garambois View Post

I am building a new exponent hoping to improve the abundance enough to get to 1 for prime numbers up to 1e9.
But the construction will take at least 60 hours.
I started Monday April 5 in the evening at 9:00 p.m.
So I won't have this exponent until late Thursday afternoon, or Friday April 9th.
I had greatly underestimated the computation time required to build the new exponent.
But this time, I can predict more precisely the end of the calculation for Tuesday, April 13.
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