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Old 2022-01-14, 02:44   #12
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Quote:
Originally Posted by sweety439 View Post
Conjecture 1 (conjecture about square numbers and odd primes): Every number which is not twice a square number (A001105) can be written as (twice a nonzero square number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, there are 47 known counterexamples, the largest known counterexample is 43358, and I conjectured that all other numbers which is not twice a square number can be written as this form.

Code:
1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358
Conjecture 2 (conjecture about triangular numbers and odd primes): Every number which is not twice a triangular number (A002378) can be written as (twice a nonzero triangular number) + (k*p), where k is 1 for odd numbers and 2 for even numbers, and p is an odd prime, there are 8 known counterexamples, the largest known counterexample is 432, and I conjectured that all other numbers which is not twice a triangular number can be written as this form.

Code:
1, 3, 4, 10, 14, 122, 422, 432
Now OEIS has sequences about these two conjectures!!! A347567 A347568
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Old 2022-01-16, 11:48   #13
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Related research: perfect power (0 and 1 are not counted as perfect power) + prime (the even prime 2 is allowed)

OEIS sequences for this: A119748 A277075 A196228 A253238 A276711

In the past, I conjectured that all integers >24 can be written as perfect power (0 and 1 are not counted as perfect power) + prime, I tested dozens of numbers, and I found many numbers having only one way to write:

Code:
6 = 4 + 2
7 = 4 + 3
9 = 4 + 5
10 = 8 + 2
12 = 9 + 3
13 = 8 + 5
14 = 9 + 5
16 = 9 + 7
17 = 4 + 13
18 = 16 + 2
20 = 9 + 11
22 = 9 + 13
25 = 8 + 17
26 = 9 + 17
31 = 8 + 23
36 = 25 + 11
42 = 25 + 17
48 = 25 + 23
58 = 27 + 31
60 = 49 + 11
64 = 27 + 37
74 = 27 + 47
76 = 9 + 67
82 = 9 + 73
85 = 32 + 53
90 = 49 + 41
114 = 25 + 89
120 = 49 + 71
127 = 125 + 2
170 = 81 + 89
193 = 36 + 157
196 = 125 + 71
202 = 9 + 193
214 = 125 + 89
324 = 125 + 199
328 = 225 + 103
331 = 324 + 7
370 = 243 + 127
412 = 81 + 331
505 = 324 + 181
562 = 225 + 337
676 = 243 + 433
706 = 243 + 463
730 = 243 + 487
799 = 576 + 223
841 = 32 + 809
1024 = 27 + 997
1087 = 36 + 1051
1204 = 81 + 1123
1243 = 324 + 919
1681 = 128 + 1553
1849 = 128 + 1721
2146 = 9 + 2137
2293 = 1296 + 997
2986 = 125 + 2861
3319 = 128 + 3191
10404 = 343 + 10061
46656 = 46225 + 431
52900 = 35937 + 16963
(I checked all numbers <= 52900, the OEIS reference shows that 112896 and 122500 are also such numbers, but not all smaller numbers are checked by me, however, due to the OEIS reference, all smaller missed numbers must be primes)

but I don't know why my conjecture fails at the number 1549, also 1771561 is another counterexample, it is known (checked by others), my conjecture works at all numbers <= 10^10 except 1549 and 1771561 (the small numbers cannot be written as this way is 1, 2, 3, 4, 5, 8, 24, thus the set of all numbers which cannot be written as this way is (likely) {1, 2, 3, 4, 5, 8, 24, 1549, 1771561}

(if only odd primes are allowed, and the even prime 2 is not allowed, then the set of all numbers is (likely) {1, 2, 3, 4, 5, 6, 8, 10, 18, 24, 127, 1549, 1771561}, the number 127 is interesting as it is the first odd number >3 which is not the sum of power of 2 and a prime)
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Old 2022-01-16, 16:55   #14
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Compare with the numbers with only one way to write as:

"twice a positive square number" + "odd prime or twice an odd prime"

or

"twice a positive triangular number" + "odd prime or twice an odd prime"

For the former (square number):

Code:
5 = 1*3 + 2*1^2
7 = 1*5 + 2*1^2
9 = 1*7 + 2*1^2
11 = 1*3 + 2*2^2
12 = 2*5 + 2*1^2
14 = 2*3 + 2*2^2
16 = 2*7 + 2*1^2
22 = 2*7 + 2*2^2
23 = 1*5 + 2*3^2
27 = 1*19 + 2*2^2
29 = 1*11 + 2*3^2
30 = 2*11 + 2*2^2
33 = 1*31 + 2*1^2
34 = 2*13 + 2*2^2
36 = 2*17 + 2*1^2
38 = 2*3 + 2*4^2
41 = 1*23 + 2*3^2
44 = 2*13 + 2*3^2
47 = 1*29 + 2*3^2
48 = 2*23 + 2*1^2
52 = 2*17 + 2*3^2
53 = 1*3 + 2*5^2
57 = 1*7 + 2*5^2
58 = 2*13 + 2*4^2
59 = 1*41 + 2*3^2
65 = 1*47 + 2*3^2
71 = 1*53 + 2*3^2
80 = 2*31 + 2*3^2
83 = 1*11 + 2*6^2
86 = 2*7 + 2*6^2
92 = 2*37 + 2*3^2
95 = 1*23 + 2*6^2
100 = 2*41 + 2*3^2
102 = 2*47 + 2*2^2
107 = 1*89 + 2*3^2
110 = 2*19 + 2*6^2
113 = 1*41 + 2*6^2
123 = 1*73 + 2*5^2
140 = 2*61 + 2*3^2
143 = 1*71 + 2*6^2
146 = 2*37 + 2*6^2
148 = 2*73 + 2*1^2
149 = 1*131 + 2*3^2
152 = 2*67 + 2*3^2
158 = 2*43 + 2*6^2
161 = 1*89 + 2*6^2
164 = 2*73 + 2*3^2
188 = 2*13 + 2*9^2
194 = 2*61 + 2*6^2
197 = 1*179 + 2*3^2
198 = 2*83 + 2*4^2
212 = 2*97 + 2*3^2
218 = 2*73 + 2*6^2
230 = 2*79 + 2*6^2
233 = 1*71 + 2*9^2
239 = 1*167 + 2*6^2
240 = 2*71 + 2*7^2
257 = 1*239 + 2*3^2
266 = 2*97 + 2*6^2
272 = 2*127 + 2*3^2
278 = 2*103 + 2*6^2
281 = 1*263 + 2*3^2
284 = 2*61 + 2*9^2
287 = 1*269 + 2*3^2
290 = 2*109 + 2*6^2
302 = 2*7 + 2*12^2
308 = 2*73 + 2*9^2
314 = 2*13 + 2*12^2
317 = 1*29 + 2*12^2
318 = 2*59 + 2*10^2
323 = 1*251 + 2*6^2
332 = 2*157 + 2*3^2
340 = 2*89 + 2*9^2
347 = 1*59 + 2*12^2
356 = 2*97 + 2*9^2
362 = 2*37 + 2*12^2
368 = 2*103 + 2*9^2
383 = 1*311 + 2*6^2
386 = 2*157 + 2*6^2
404 = 2*193 + 2*3^2
407 = 1*389 + 2*3^2
410 = 2*61 + 2*12^2
413 = 1*251 + 2*9^2
422 = 2*67 + 2*12^2
438 = 2*23 + 2*14^2
442 = 2*157 + 2*8^2
443 = 1*281 + 2*9^2
446 = 2*79 + 2*12^2
449 = 1*431 + 2*3^2
458 = 2*193 + 2*6^2
470 = 2*199 + 2*6^2
482 = 2*97 + 2*12^2
492 = 2*197 + 2*7^2
500 = 2*241 + 2*3^2
506 = 2*109 + 2*12^2
530 = 2*229 + 2*6^2
536 = 2*43 + 2*15^2
542 = 2*127 + 2*12^2
548 = 2*193 + 2*9^2
554 = 2*241 + 2*6^2
566 = 2*139 + 2*12^2
569 = 1*281 + 2*12^2
590 = 2*151 + 2*12^2
596 = 2*73 + 2*15^2
602 = 2*157 + 2*12^2
620 = 2*229 + 2*9^2
626 = 2*277 + 2*6^2
632 = 2*307 + 2*3^2
638 = 2*283 + 2*6^2
650 = 2*181 + 2*12^2
656 = 2*103 + 2*15^2
662 = 2*7 + 2*18^2
668 = 2*109 + 2*15^2
680 = 2*331 + 2*3^2
692 = 2*337 + 2*3^2
698 = 2*313 + 2*6^2
743 = 1*293 + 2*15^2
773 = 1*701 + 2*6^2
782 = 2*67 + 2*18^2
785 = 1*137 + 2*18^2
788 = 2*313 + 2*9^2
794 = 2*73 + 2*18^2
798 = 2*383 + 2*4^2
818 = 2*373 + 2*6^2
824 = 2*331 + 2*9^2
848 = 2*199 + 2*15^2
863 = 1*701 + 2*9^2
872 = 2*211 + 2*15^2
884 = 2*433 + 2*3^2
890 = 2*409 + 2*6^2
926 = 2*139 + 2*18^2
938 = 2*433 + 2*6^2
980 = 2*409 + 2*9^2
998 = 2*463 + 2*6^2
1010 = 2*181 + 2*18^2
1022 = 2*367 + 2*12^2
1082 = 2*397 + 2*12^2
1094 = 2*223 + 2*18^2
1118 = 2*523 + 2*6^2
1124 = 2*337 + 2*15^2
1148 = 2*349 + 2*15^2
1172 = 2*577 + 2*3^2
1178 = 2*13 + 2*24^2
1220 = 2*601 + 2*3^2
1227 = 1*1129 + 2*7^2
1232 = 2*607 + 2*3^2
1238 = 2*43 + 2*24^2
1292 = 2*421 + 2*15^2
1322 = 2*337 + 2*18^2
1367 = 1*719 + 2*18^2
1388 = 2*613 + 2*9^2
1415 = 1*263 + 2*24^2
1418 = 2*673 + 2*6^2
1478 = 2*163 + 2*24^2
1502 = 2*607 + 2*12^2
1562 = 2*457 + 2*18^2
1586 = 2*757 + 2*6^2
1598 = 2*223 + 2*24^2
1622 = 2*487 + 2*18^2
1668 = 2*809 + 2*5^2
1670 = 2*691 + 2*12^2
1703 = 1*821 + 2*21^2
1748 = 2*433 + 2*21^2
1754 = 2*733 + 2*12^2
1787 = 1*1499 + 2*12^2
1828 = 2*73 + 2*29^2
1844 = 2*193 + 2*27^2
1892 = 2*937 + 2*3^2
1898 = 2*373 + 2*24^2
1940 = 2*241 + 2*27^2
1958 = 2*79 + 2*30^2
1988 = 2*769 + 2*15^2
2042 = 2*877 + 2*12^2
2060 = 2*1021 + 2*3^2
2090 = 2*1009 + 2*6^2
2123 = 1*971 + 2*24^2
2132 = 2*337 + 2*27^2
2138 = 2*1033 + 2*6^2
2174 = 2*1051 + 2*6^2
2180 = 2*1009 + 2*9^2
2210 = 2*1069 + 2*6^2
2234 = 2*541 + 2*24^2
2328 = 2*1163 + 2*1^2
2342 = 2*271 + 2*30^2
2402 = 2*877 + 2*18^2
2408 = 2*1123 + 2*9^2
2438 = 2*643 + 2*24^2
2486 = 2*919 + 2*18^2
2507 = 1*1049 + 2*27^2
2558 = 2*379 + 2*30^2
2582 = 2*967 + 2*18^2
2648 = 2*883 + 2*21^2
2708 = 2*1129 + 2*15^2
2732 = 2*277 + 2*33^2
2762 = 2*1237 + 2*12^2
2768 = 2*1303 + 2*9^2
2822 = 2*1087 + 2*18^2
2858 = 2*853 + 2*24^2
2900 = 2*1009 + 2*21^2
2933 = 1*2861 + 2*6^2
3002 = 2*601 + 2*30^2
3062 = 2*631 + 2*30^2
3110 = 2*1231 + 2*18^2
3242 = 2*1297 + 2*18^2
3284 = 2*1201 + 2*21^2
3317 = 1*3299 + 2*3^2
3434 = 2*421 + 2*36^2
3452 = 2*997 + 2*27^2
3482 = 2*1597 + 2*12^2
3515 = 1*2633 + 2*21^2
3530 = 2*1621 + 2*12^2
3572 = 2*1777 + 2*3^2
3620 = 2*1801 + 2*3^2
3662 = 2*67 + 2*42^2
3713 = 1*1913 + 2*30^2
3722 = 2*97 + 2*42^2
3758 = 2*1303 + 2*24^2
3770 = 2*1741 + 2*12^2
3962 = 2*1657 + 2*18^2
3980 = 2*1549 + 2*21^2
3998 = 2*1423 + 2*24^2
4022 = 2*1867 + 2*12^2
4082 = 2*277 + 2*42^2
4118 = 2*1483 + 2*24^2
4148 = 2*1993 + 2*9^2
4178 = 2*2053 + 2*6^2
4292 = 2*2137 + 2*3^2
4334 = 2*2131 + 2*6^2
4490 = 2*1669 + 2*24^2
4502 = 2*487 + 2*42^2
4532 = 2*241 + 2*45^2
4538 = 2*1693 + 2*24^2
4568 = 2*2203 + 2*9^2
4586 = 2*997 + 2*36^2
4673 = 1*2081 + 2*36^2
4688 = 2*823 + 2*39^2
4820 = 2*1321 + 2*33^2
4832 = 2*1327 + 2*33^2
4958 = 2*1579 + 2*30^2
5078 = 2*2503 + 2*6^2
5102 = 2*787 + 2*42^2
5300 = 2*1129 + 2*39^2
5612 = 2*2797 + 2*3^2
5642 = 2*2677 + 2*12^2
5708 = 2*829 + 2*45^2
5798 = 2*1999 + 2*30^2
5852 = 2*2917 + 2*3^2
5942 = 2*2647 + 2*18^2
5987 = 1*2459 + 2*42^2
6008 = 2*1483 + 2*39^2
6188 = 2*1069 + 2*45^2
6218 = 2*193 + 2*54^2
6302 = 2*2251 + 2*30^2
6332 = 2*2437 + 2*27^2
6368 = 2*1663 + 2*39^2
6518 = 2*2683 + 2*24^2
6602 = 2*997 + 2*48^2
6797 = 1*6779 + 2*3^2
6836 = 2*2689 + 2*27^2
6938 = 2*3433 + 2*6^2
7004 = 2*3061 + 2*21^2
7142 = 2*2671 + 2*30^2
7622 = 2*211 + 2*60^2
7718 = 2*3823 + 2*6^2
7730 = 2*3541 + 2*18^2
7928 = 2*3739 + 2*15^2
7982 = 2*3847 + 2*12^2
8432 = 2*967 + 2*57^2
8444 = 2*1621 + 2*51^2
8558 = 2*4243 + 2*6^2
8660 = 2*3889 + 2*21^2
8828 = 2*2389 + 2*45^2
9008 = 2*4423 + 2*9^2
9020 = 2*541 + 2*63^2
9122 = 2*2797 + 2*42^2
9290 = 2*2341 + 2*48^2
9308 = 2*2053 + 2*51^2
9422 = 2*4567 + 2*12^2
9722 = 2*2557 + 2*48^2
9860 = 2*4201 + 2*27^2
10964 = 2*3457 + 2*45^2
11012 = 2*5281 + 2*15^2
11090 = 2*4969 + 2*24^2
11498 = 2*2833 + 2*54^2
11972 = 2*2017 + 2*63^2
12062 = 2*3727 + 2*48^2
12098 = 2*1693 + 2*66^2
12548 = 2*3673 + 2*51^2
12602 = 2*1117 + 2*72^2
12878 = 2*2083 + 2*66^2
14018 = 2*4093 + 2*54^2
14162 = 2*997 + 2*78^2
14882 = 2*7297 + 2*12^2
15758 = 2*823 + 2*84^2
15908 = 2*7873 + 2*9^2
16172 = 2*6997 + 2*33^2
16838 = 2*5503 + 2*54^2
17168 = 2*3823 + 2*69^2
17648 = 2*8599 + 2*15^2
18428 = 2*9133 + 2*9^2
19142 = 2*1471 + 2*90^2
20330 = 2*3109 + 2*84^2
20918 = 2*9883 + 2*24^2
21548 = 2*10333 + 2*21^2
21722 = 2*457 + 2*102^2
23018 = 2*2293 + 2*96^2
23612 = 2*9781 + 2*45^2
25022 = 2*6427 + 2*78^2
27668 = 2*8209 + 2*75^2
30212 = 2*7537 + 2*87^2
30668 = 2*13309 + 2*45^2
31130 = 2*15241 + 2*18^2
32162 = 2*15937 + 2*12^2
32372 = 2*7537 + 2*93^2
47702 = 2*6427 + 2*132^2
63758 = 2*28279 + 2*60^2
66410 = 2*6961 + 2*162^2
89072 = 2*11047 + 2*183^2
For the latter (triangular number):

Code:
5 = 1*3 + 1*2
7 = 1*5 + 1*2
8 = 2*3 + 1*2
11 = 1*5 + 2*3
18 = 2*3 + 3*4
21 = 1*19 + 1*2
22 = 2*5 + 3*4
24 = 2*11 + 1*2
27 = 1*7 + 4*5
32 = 2*13 + 2*3
38 = 2*13 + 3*4
50 = 2*19 + 3*4
51 = 1*31 + 4*5
54 = 2*17 + 4*5
57 = 1*37 + 4*5
60 = 2*29 + 1*2
62 = 2*3 + 7*8
74 = 2*31 + 3*4
84 = 2*41 + 1*2
105 = 1*103 + 1*2
108 = 2*53 + 1*2
111 = 1*109 + 1*2
126 = 2*53 + 4*5
140 = 2*67 + 2*3
150 = 2*47 + 7*8
174 = 2*59 + 7*8
180 = 2*89 + 1*2
186 = 2*83 + 4*5
242 = 2*43 + 12*13
252 = 2*71 + 10*11
258 = 2*101 + 7*8
270 = 2*107 + 7*8
357 = 1*337 + 4*5
372 = 2*131 + 10*11
471 = 1*199 + 16*17
492 = 2*191 + 10*11
510 = 2*227 + 7*8
630 = 2*179 + 16*17
666 = 2*197 + 16*17
690 = 2*317 + 7*8
765 = 1*709 + 7*8
792 = 2*71 + 25*26
810 = 2*269 + 16*17
1080 = 2*449 + 13*14
1112 = 2*541 + 5*6
1380 = 2*599 + 13*14
1434 = 2*311 + 28*29
1602 = 2*773 + 7*8
1848 = 2*599 + 25*26
1920 = 2*257 + 37*38
2160 = 2*827 + 22*23
3726 = 2*1367 + 31*32
4752 = 2*1151 + 49*50
5397 = 1*1237 + 64*65
5652 = 2*1601 + 49*50
7800 = 2*2819 + 46*47
12420 = 2*5507 + 37*38
16632 = 2*3851 + 94*95
Like "perfect power (not including 0 and 1) + prime", it is conjectured that 89072 and 16632 are the largest examples of them, respectively.
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Old 2022-01-16, 17:05   #15
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There are complement of A065377 and A065397 of the primes A000040 in OEIS: A065376 and A065396, respectively.

Last fiddled with by sweety439 on 2022-01-16 at 17:06
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Old 2022-01-20, 22:18   #16
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"99(4^34019)99 palind"
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The conjecture 2 in post #1 (i.e. A347568 is full with 8 terms: {1, 3, 4, 10, 14, 122, 422, 432}):

Zhi-Wei Sun, Conjectures on sums of primes and triangular numbers

(page 3 has "furthermore, any positive integer n 6∈ {2, 5, 7, 61, 211, 216} can be written in the form p + Tx with x ∈ Z+, where p is an odd prime or zero", which is equivalent to the only even numbers in A347568 are {4, 10, 14, 122, 422, 432}) and (page 5 has "Any odd integer n > 3 can be written in the form p + x(x + 1) with p a prime and and x a positive integer", which is equivalent to the only odd numbers in A347568 are {1, 3})

The conjecture 1 in post #1 (i.e. A347567 is full with 47 terms: {1, 3, 4, 6, 10, 17, 20, 26, 62, 68, 74, 116, 122, 137, 170, 182, 227, 254, 260, 428, 452, 740, 758, 878, 977, 1052, 1142, 1187, 1412, 1460, 1493, 1542, 1658, 1982, 2510, 2702, 2828, 3098, 3812, 5777, 5972, 5993, 6638, 7352, 15098, 19268, 43358}):

G. H. Hardy, J. E. Littlewood, Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes (see the attached pdf file)

(page 49 has "probably because, the idea that every number is a square,
or the sum of a prime and a square, is refuted (even if I is counted as a prime)
by such immediate examples as 34 and 58. But the problem of the representation
of an odd number in the form t+ 2m^2 has received some attention; and
it has been verified that the only odd numbers less than 9000, and not of the
form desired, are 5 777 and 5 993", which is related to the even numbers in A347567 and the odd numbers in A347567, respectively)
Attached Files
File Type: pdf BF02403921.pdf (2.47 MB, 2 views)
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∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔