(conjectured or proved) finite-record sequences

sweety439 · 2022-03-06, 12:23

only list the base-independent sequences, thus for example, least start of exactly n consecutive numbers that are harshad numbers (A060159) is not listed here.

A343816: least start of exactly n consecutive odd numbers that are cyclic numbers: {23, 41, 177, 1, 11, 877, 2387, 695} (no further terms, since numbers divisible by 9 cannot be cyclic number)

A138496: record number of occurrences of n as an entry in rows <= n of Pascal's triangle: {0, 1, 10, 120, 3003}, or {2, 3, 6, 10, 120, 3003} (if the number 1 (which appear infinitely many times in Pascal's triangle) is not counted, see A062527), is there a number (>1) appearing >8 times in Pascal's triangle?

{6, 13, 454} with difference {25-6=19, 35-27=8, 602-596=6} (see A152454, record with smaller difference in a single row of A152454), is there a next row with a smaller difference?

{0, 2, 383, 10103}, see A000790, with record values {4, 341, 382, 561}, also for the negative bases, {0, 1, 13, 22, 58, 277, 877, 1822, 5098, 23602}, with record values {4, 9, 14, 21, 57, 69, 219, 451, 481, 561} (no further terms, since 561 is a Carmichael number, thus 561 is weak pseudoprime to every (positive or negative or 0) base)

Record of the smallest n>=1 such that k*2^n+1 is prime for odd k: {1, 7, 17, 19, 31, 47, 383, 2897, 3061, 4847, 5359, 10223, 21181, ..., 78557}, the record values are {1, 2, 3, 6, 8, 583, 6393, 9715, 33288, 3321063, 5054502, 31172165, ..., infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 78557*2^n+1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

Record of the smallest n>=1 such that 2^n+k is prime (or PRP) for odd k: {1, 7, 23, 31, 47, 61, 139, 271, 287, 773, 2131, 40291, 78557}, the record values are {1, 2, 3, 4, 5, 8, 10, 20, 29, 955, 4583176, 9092392, infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 2^n+78557 is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

Record of the smallest n>=1 such that k*2^n-1 is prime for odd k: {1, 13, 23, 43, 59, 127, 191, 659, 2293, 23669, ..., 509203}, the record values are {2, 3, 4, 7, 12, 25, 226, 800516, 12918431, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 509203*2^n-1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

Record of the smallest n>=1 such that 2^n-k is prime (or PRP) for odd k: {1, 3, 7, 127, 221, 337, 1061, 1871, ..., 509203}, the record values are {2, 3, 39, 47, 714, 791, 970, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 2^n-509203 is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

Record of the smallest n>=1 such that |2^n-k| is prime (or PRP) for odd k: {1, 3, 29, 127, 337, 2293, ..., 509203}, the record values are {2, 3, 4, 47, 791, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that |2^n-509203| is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

Primes p such that the absolute value of the fraction A260209(A000720(p)) / (p^3) is a record low: {2, 3, 11, 13, 31, 103, 211, 2851, 12101, 16843}, or start with the prime 5, {5, 11, 13, 31, 103, 211, 2851, 12101, 16843} (no further terms, since for the prime 16843, the value is 0)

sweety439 · 2022-03-06, 13:44

A291466: least start of exactly n consecutive numbers that are sphenic numbers: {30, 230, 1309} (no further terms, since numbers divisible by 4 cannot be sphenic numbers)

least start of exactly n consecutive numbers that are Achilles numbers: {72, 5425069447,

{105, 66, 406} least start of exactly n consecutive triangular numbers that are sphenic numbers (no further terms, since if 4 consecutive triangular numbers are all sphenic numbers, let them be n*(n+1)/2, (n+1)*(n+2)/2, (n+2)*(n+3)/2, (n+3)*(n+4)/2, then we have omega(n*(n+1)) = omega((n+1)*(n+2)) = omega((n+2)*(n+3)) * omega((n+3)*(n+4) = 4, and none of n, n+1, n+2, n+3, n+4 is divisible by 8 (or at least one of n*(n+1)/2, (n+1)*(n+2)/2, (n+2)*(n+3)/2, (n+3)*(n+4)/2 will be divisible by 4 and cannot be sphenic number), thus omega(n) = omega(n+2) = omega(n+4), but n, n+2, n+4 cannot be all primes unless n=3, thus omega(n) must be >= 2, and if omega(n) = 2, then we have omega(n+1) = 2 (since omega(n*(n+1)) = 4), similarly, omega(n+2) = omega(n+3) = omega(n+4) = 2, which is impossible since at least one of n, n+1, n+2, n+3, n+4 is divisible by 4, thus this number can only be 4, thus omega(n) must be 3, and omega(n+1) = omega(n+3) = 1, i.e. n+1 and n+3 are twin primes, thus neither n+1 nor n+3 is divisible by 3, thus n+2 is divisible by 3, and n+2 cannot be divisible by 12 since omega(n+2) = 3 and if n+2 is divisible by 12 then n+2 can only be 12, thus n and n+4 must be divisible by 4 (note that neither n+1 nor n+3 cannot be divisible by 4 since n+1 and n+3 are twin primes), and thus one of n and n+4 will be divisible by 8, which is a contradiction!

The smallest prime factor of A005165(n), this record sequence is finite since for all n>=3612702, the smallest prime factor of A005165(n) are all 3612703, but we do not know the last record n and its record value (since this requires to factor near 22122513-digit numbers, the last record n and its record value will not be known in our lifetime! but we know that they really exist! this is equivalent to this problem: Find the smallest m such that gcd(A005165(n),m) > 1 for all n >= 2, such m must exist, but we cannot find the m in our lifetime)

sweety439 · 2022-04-24, 20:29

Related page: What is special about this number? (Some numbers, like 43 and 94, breaks a law of small numbers, also some numbers, like 24 and 41, are the (proved) largest number satisfying a property.

sweety439 · 2022-05-14, 13:36

Quote:

Originally Posted by sweety439

Record of the smallest n>=1 such that k*2^n+1 is prime for odd k: {1, 7, 17, 19, 31, 47, 383, 2897, 3061, 4847, 5359, 10223, 21181, ..., 78557}, the record values are {1, 2, 3, 6, 8, 583, 6393, 9715, 33288, 3321063, 5054502, 31172165, ..., infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 78557*2^n+1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

Record of the smallest n>=1 such that 2^n+k is prime (or PRP) for odd k: {1, 7, 23, 31, 47, 61, 139, 271, 287, 773, 2131, 40291, 78557}, the record values are {1, 2, 3, 4, 5, 8, 10, 20, 29, 955, 4583176, 9092392, infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 2^n+78557 is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

Record of the smallest n>=1 such that k*2^n-1 is prime for odd k: {1, 13, 23, 43, 59, 127, 191, 659, 2293, 23669, ..., 509203}, the record values are {2, 3, 4, 7, 12, 25, 226, 800516, 12918431, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 509203*2^n-1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

Record of the smallest n>=1 such that 2^n-k is prime (or PRP) for odd k: {1, 3, 7, 127, 221, 337, 1061, 1871, ..., 509203}, the record values are {2, 3, 39, 47, 714, 791, 970, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 2^n-509203 is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

Record of the smallest n>=1 such that |2^n-k| is prime (or PRP) for odd k: {1, 3, 29, 127, 337, 2293, ..., 509203}, the record values are {2, 3, 4, 47, 791, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that |2^n-509203| is prime (or we can say this n is infinity), thus of course there cannot be any further terms.

We use "infinity" instead of "0" or "-1" because ...

* infinity is > any finite number, this is more convenient (e.g. the smallest n such that (k^n-1)/(k-1) is prime, for k = 185, (k^n-1)/(k-1) has been searched to n=66337 with no prime or PRP found, we can use ">66337" for k = 185, ">66337" includes infinity (since infinity is > 66337) but does not includes 0 or -1, it is still possible that there is no n such that (k^n-1)/(k-1) is prime for k = 185)
* "the smallest n such that (k^n-1)/(k-1) is prime" is "the largest n such that (k^r-1)/(k-1) is not prime for all r<n", for the case that (k^n-1)/(k-1) is never prime (k = 9, 25, 32, 49, 64, 81, 121, 125, 144, ...), such n is infinity instead of 0 or -1
* let S(k) = {n: (k^n-1)/(k-1) is prime}, a(n) = min(S(k)), then a(n) = the smallest n such that (k^n-1)/(k-1) is prime, and S(2) = {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, ...}, a(2) = 2, S(3) = {3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, ...}, a(3) = 3, S(8) = {3}, a(8) = 3, S(9) = {} (the empty set), a(9) = infinity (due to the definition of "sup" and "inf", sup(empty set) = -infinity and inf(empty set) = +infinity) instead of 0 or -1
* A058887(n) is the smallest k such that A057192(k) > n, A058887(n) <= 271129 for all n, and A057192(271129) should be infinity instead of 0 or -1

(but the only technical problem is that OEIS sequences cannot use "infinity" thus in OEIS sequence we can use "0" or "-1" to replace "infinity", such as A084740, a(9), a(25), a(32), a(49), a(64), etc. should be "infinity" instead of "0", also A040076(78557) and A040081(509203), should be "infinity" instead of "-1", but an example is A001438, which a(0) = a(1) = infinity

Also see http://gladhoboexpress.blogspot.com/...-derbread.html and http://chesswanks.com/seq/a306861.txt and http://chesswanks.com/seq/a269254.txt

sweety439 · 2022-12-08, 23:27

Quote:

Originally Posted by sweety439

We use "infinity" instead of "0" or "-1" because ...

Another example of "infinity" in number theory:

Since for all positive integer n we have 1^(p-1) == 1 mod p^n

Code:

 Basis  Primzahlen^2 (P^2)
    1   All primes^infinity
    2   1093, 3511
    3   11, 1006003
    4   1093, 3511
    5   2, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
    6   66161, 534851, 3152573
    7   5, 491531
    8   3, 1093, 3511
    9   2^3, 11, 1006003
   10   3, 487, 56598313
   11   71
   12   2693, 123653
   13   2, 863, 1747591
   14   29, 353, 7596952219
   15   29131, 119327070011
   16   1093, 3511
   17   2^4, 3, 46021, 48947, 478225523351
   18   5, 7^3, 37, 331, 33923, 1284043
   19   3, 7^3, 13, 43, 137, 63061489
   20   281, 46457, 9377747, 122959073
   21   2
   22   13, 673, 1595813, 492366587
   23   13, 2481757, 13703077, 15546404183
   24   5, 25633
   25   2^3, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
   26   3^3, 5, 71, 486999673, 6695256707
   27   11, 1006003
   28   3^3, 19, 23
   29   2
   30   7, 160541, 94727075783
   31   7, 79, 6451, 2806861
   32   5, 1093, 3511
   33   2^5, 233, 47441, 9639595369
   34   46145917691
   35   3, 1613, 3571
   36   66161, 534851, 3152573
   37   2, 3, 77867, 76407520781
   38   17, 127
   39   8039
   40   11, 17, 307, 66431
   41   2^3, 29, 1025273, 138200401
   42   23^3, 719867822369
   43   5, 103
   44   3, 229, 5851
   45   2, 1283, 131759, 157635607
   46   3, 829
   47  
   48   7, 257
   49   2^4, 5, 491531
   50   7
   51   5, 41
   52   461, 1228488439
   53   2, 3^3, 47, 59, 97
   54   19, 1949
   55   3^3, 30109, 7278001, 27207490529, 902060958301
   56   647, 7079771, 115755260963
   57   2^3, 5^3, 47699, 86197
   58   131, 42250279
   59   2777
   60   29, 9566295763
   61   2
   62   3, 19, 127, 1291
   63   23, 29, 36713, 401771
   64   3, 1093, 3511
   65   2^6, 17, 163
   66   89351671, 588024812497
   67   7, 47, 268573
   68   5^3, 7, 19, 113^3, 2741
   69   2, 19, 223, 631, 2503037
   70   13, 142963
   71   3, 47, 331
   72  
   73   2^3, 3
   74   5
   75   17, 43, 347, 31247
   76   5, 37, 1109, 9241, 661049, 20724663983
   77   2, 32687
   78   43, 151, 181, 1163, 56149, 4229335793
   79   7, 263, 3037, 1012573, 60312841
   80   3^4, 7, 13, 6343
   81   2^4, 11, 1006003
   82   3^4, 5
   83   4871, 13691, 315746063
   84   163, 653, 20101, 663840051067
   85   2, 11779
   86   68239, 6232426549
   87   1999, 48121, 604807523183
   88   2535619637
   89   2^3, 3, 13
   90   6590291053
   91   3, 293
   92   727, 383951, 12026117, 18768727, 1485161969
   93   2, 5, 509, 9221, 81551
   94   11, 241, 32143, 463033
   95   2137, 15061, 96185643031
   96   109, 5437, 8329, 12925267, 103336004179
   97   2^5, 7, 2914393, 76704103313
   98   3, 28627, 61001527
   99   5, 7, 13, 19, 83
  100   3, 487, 56598313
  101   2, 5, 1050139
  102   7559, 11813, 139409857
  103   24490789
  104   313, 237977, 11950711691
  105   2^3, 7669
  106   79399, 672799
  107   3^3, 5, 97, 613181
  108   3761, 10271, 1296018233
  109   2, 3^3, 20252173
  110   17, 5381, 9431
  111   131
  112   11, 1037888513
  113   2^4, 405697846751
  114   9181
  115   31, 2743780307
  116   3, 7, 19, 47
  117   2, 7, 31, 37, 182111
  118   3, 5, 11, 23, 3152249, 10404887
  119   1741
  120   11, 653, 2074031, 124148023
  121   2^3, 71
  122   11, 2791, 16522744709
  123   34849
  124   5^3, 11^3, 22511
  125   2, 3, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801
  126   5^3
  127   3, 19, 907, 13778951
  128   7, 1093, 3511
  129   2^7, 7, 113
  130   11, 23, 62351, 70439
  131   17, 754480919
  132   5
  133   2, 5277179
  134   3^3, 17, 406531, 73629977
  135   23147324413
  136   3^3, 5153, 87249620417
  137   2^3, 29, 59, 6733, 18951271, 4483681903
  138   97, 4889
  139  
  140   14591
  141   2, 192047, 42039743
  142   143111, 12838109
  143   3, 5, 67, 197, 1999, 12848347, 65451824959, 225413072431
  144   2693, 123653
  145   2^4, 3, 31
  146   7, 13, 79, 22639
  147   13, 79, 103, 283, 24203, 4058948753
  148   7, 11, 41, 14879, 161141, 2805469039
  149   2, 5, 29573, 121456243, 2283131621
  150   13, 21199069, 85877201, 103053130679
  151   5, 2251, 14107, 5288341, 15697215641
  152   3, 22554863, 54504719
  153   2^3, 8539, 5540099
  154   3
  155   17, 211691, 309131, 6282943, 1196757167
  156   347, 1297
  157   2, 5, 122327, 4242923, 5857727461, 275318049829
  158   17^3, 1031, 820279
  159   367, 5356051
  160   72669541193
  161   2^5, 3^4, 11^3, 83, 137, 6188401, 93405163
  162   607, 142946059423
  163   3^4, 3898031
  164   103, 107, 1493
  165   2, 7, 211, 25102757, 11699444843
  166   7, 877, 61819
  167   64661497
  168   5, 13, 739, 27427
  169   2^3, 863, 1747591
  170   3, 13, 23, 1163, 170613512651
  171   269, 1803383
  172   3, 47, 91303
  173   2, 3079, 56087
  174   5, 113, 199, 379
  175   397, 487, 2199439603
  176   5, 191, 5611481459
  177   2^4, 7, 23, 42397, 123681208241
  178   7, 1213, 67751, 296036443
  179   3, 17, 35059, 126443
  180   269, 5737
  181   2, 3, 101
  182   5^4, 211, 8934721
  183   1231487
  184   89, 661, 1187, 2203, 6043
  185   2^3, 12577
  186  
  187  
  188   3^3, 13
  189   2, 1847, 6199
  190   3^3, 29, 236981, 382351, 753743
  191   13, 379133
  192   13, 4049, 302279, 465435739
  193   2^6, 5^3, 4877
  194   439, 8338826207
  195   7, 23, 13421, 3132631
  196   29, 353, 7596952219
  197   2, 3, 7, 653, 6237773
  198   147647, 545639981
  199   3, 5, 77263, 1843757
  200  
  201   2^3, 5, 86722091
  202   11, 47
  203  
  204   5431, 142061, 1070347, 15028133
  205   2, 367
  206   3, 151, 1201, 2087
  207   5, 41, 269, 809, 26240063
  208   3, 947
  209   2^4, 229, 2203, 511654217, 6802750783
  210   43, 449
  211   279311
  212   1960573
  213   2, 313, 1877
  214   7, 17
  215   3^3, 7, 11, 3327697, 502095466241
  216   66161, 534851, 3152573
  217   2^3, 3^3, 127447, 8468323
  218   5, 163
  219   7891033151
  220   3581, 9619, 79867, 717967, 13477271, 32376410527, 34816906301
  221   2, 29, 1669, 15233, 103361803, 406833359, 455317162337, 1180548631831
  222  
  223   71, 349
  224   3, 5, 17, 130677149
  225   2^5, 29131, 119327070011
  226   3, 5, 7, 97, 157
  227   7, 40277
  228   1361, 182353
  229   2, 31
  230   47, 384533887
  231  
  232   5, 30851, 838667, 2250741293
  233   2^3, 3, 11, 157, 86735239
  234   19, 46441, 817766595407
  235   3, 31, 118022689
  236   29, 229
  237   2
  238   977, 8609, 4997219, 863039377, 64904716187
  239   11, 13^4, 74047, 212855197, 361552687, 12502228667
  240   227797
  241   2^4, 11, 523, 1163, 35407
  242   3^5, 373, 2386301, 2434609, 167962252433
  243   5, 11, 1006003
  244   3^5, 7, 16906680221
  245   2, 11, 19, 229, 218173429
  246   7, 877441867
  247   5081, 105527103139
  248   67, 101, 663893, 502608046943, 668677573093
  249   2^3, 5^3, 13, 17
  250   131, 509
  251   3, 5^3, 11, 17, 421, 395696461
  252   211, 997, 3613, 394213789
  253   2, 3, 167, 90901, 2905031
  254   89, 2269, 18042251, 32984389
  255   23, 463
  256   1093, 3511
  257   2^8, 5, 359, 49559, 648258371
  258   13, 17827
  259   173
  260   3, 83, 15263, 27827, 1749229, 6647339, 19148231, 3977977109
  261   2, 43, 71, 257
  262   3, 19, 571
  263   7, 23, 251, 267541, 159838801
  264   7, 61, 176051, 1446587
  265   2^3, 10853
  266   23, 163, 21401
  267   29, 20411, 46926349
  268   5, 13, 101
  269   2, 3^3, 11, 83, 8779, 65684482177
  270   29
  271   3^3, 168629, 16774141, 235558417, 12145092821
  272   3833, 14084849
  273   2^4, 1086731
  274   5, 23, 16631
  275   7, 6323, 134513, 138641, 455471, 40196043527, 279203122727
  276   5, 7, 43, 193
  277   2, 1993, 243547988443
  278   3, 1741
  279   67, 97, 557142359
  280   3, 47, 401
  281   2^3, 3443059
  282   5, 11, 13120561
  283   46301
  284   197, 1103, 9127, 16189, 1601147, 10723208449
  285   2, 5791, 157907
  286   71, 619, 15017, 1016689
  287   3, 653, 967, 10322313269, 405469792823
  288   17, 43, 36979, 376721, 670507
  289   2^5, 3, 46021, 48947, 478225523351
  290   17, 6659, 319811, 273354437
  291   109, 16766930077
  292   19, 43, 887, 941, 3239857
  293   2, 5, 7, 19, 83
  294   356135477, 592538407, 367003576739
  295   7, 47^3, 26249
  296   3^3, 971
  297   2^3, 2480497, 2779225651
  298   3^3, 59, 101, 181, 598421
  299   5, 19, 59, 374893957
  300   37, 59, 29599

preda · 2022-12-09, 07:24

Quote:

Originally Posted by sweety439

Code:

 Basis  Primzahlen^2 (P^2)
    1   All primes^infinity
    2   1093, 3511
    3   11, 1006003
    4   1093, 3511
...

what is in that list?

S485122 · 2022-12-09, 09:25

Quote:

Originally Posted by sweety439

...
Since for all positive integer n we have 1^(p-1) == 1 mod p^n
...

Couldn't you simplify that :

"Since for any integers n and p, with 0<n and 1<p, we have 1 == 1 mod p^n"

or :

"Since 1=1"

or even drop that part altogether because it is always true.

kruoli · 2022-12-09, 10:05

Quote:

Originally Posted by preda

what is in that list?

It looks like numbers, where \(b^{p-1} \equiv 1 \mod p^2\) where \(b\) is "Basis" and \(p\) is "Primzahlen".

sweety439 · 2022-12-10, 06:03

Quote:

Originally Posted by preda

what is in that list?

They are the primes p such that b^(p-1) == 1 mod p^2 (called "generalized Wieferich primes base b"), if "p^3" or "p^4" are written, then this means b^(p-1) == 1 mod p^3 (or p^4), e.g. b = 18 has 7^3, and this means 18^(7-1) == 1 mod 7^3

This is the full list for bases 2 <= b <= 10000 (not include b = 1)

2022-03-06, 12:23	#1
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×7×263 Posts	(conjectured or proved) finite-record sequences only list the base-independent sequences, thus for example, least start of exactly n consecutive numbers that are harshad numbers (A060159) is not listed here. A343816: least start of exactly n consecutive odd numbers that are cyclic numbers: {23, 41, 177, 1, 11, 877, 2387, 695} (no further terms, since numbers divisible by 9 cannot be cyclic number) A138496: record number of occurrences of n as an entry in rows <= n of Pascal's triangle: {0, 1, 10, 120, 3003}, or {2, 3, 6, 10, 120, 3003} (if the number 1 (which appear infinitely many times in Pascal's triangle) is not counted, see A062527), is there a number (>1) appearing >8 times in Pascal's triangle? {6, 13, 454} with difference {25-6=19, 35-27=8, 602-596=6} (see A152454, record with smaller difference in a single row of A152454), is there a next row with a smaller difference? {0, 2, 383, 10103}, see A000790, with record values {4, 341, 382, 561}, also for the negative bases, {0, 1, 13, 22, 58, 277, 877, 1822, 5098, 23602}, with record values {4, 9, 14, 21, 57, 69, 219, 451, 481, 561} (no further terms, since 561 is a Carmichael number, thus 561 is weak pseudoprime to every (positive or negative or 0) base) Record of the smallest n>=1 such that k2^n+1 is prime for odd k: {1, 7, 17, 19, 31, 47, 383, 2897, 3061, 4847, 5359, 10223, 21181, ..., 78557}, the record values are {1, 2, 3, 6, 8, 583, 6393, 9715, 33288, 3321063, 5054502, 31172165, ..., infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 785572^n+1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that 2^n+k is prime (or PRP) for odd k: {1, 7, 23, 31, 47, 61, 139, 271, 287, 773, 2131, 40291, 78557}, the record values are {1, 2, 3, 4, 5, 8, 10, 20, 29, 955, 4583176, 9092392, infinity}, since 78557 is Sierpinski number, there is no n>=1 such that 2^n+78557 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that k2^n-1 is prime for odd k: {1, 13, 23, 43, 59, 127, 191, 659, 2293, 23669, ..., 509203}, the record values are {2, 3, 4, 7, 12, 25, 226, 800516, 12918431, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 5092032^n-1 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that 2^n-k is prime (or PRP) for odd k: {1, 3, 7, 127, 221, 337, 1061, 1871, ..., 509203}, the record values are {2, 3, 39, 47, 714, 791, 970, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that 2^n-509203 is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Record of the smallest n>=1 such that \|2^n-k\| is prime (or PRP) for odd k: {1, 3, 29, 127, 337, 2293, ..., 509203}, the record values are {2, 3, 4, 47, 791, ..., infinity}, since 509203 is Riesel number, there is no n>=1 such that \|2^n-509203\| is prime (or we can say this n is infinity), thus of course there cannot be any further terms. Primes p such that the absolute value of the fraction A260209(A000720(p)) / (p^3) is a record low: {2, 3, 11, 13, 31, 103, 211, 2851, 12101, 16843}, or start with the prime 5, {5, 11, 13, 31, 103, 211, 2851, 12101, 16843} (no further terms, since for the prime 16843, the value is 0) Last fiddled with by sweety439 on 2022-05-04 at 02:35

2022-03-06, 13:44	#2
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2×7×263 Posts	A291466: least start of exactly n consecutive numbers that are sphenic numbers: {30, 230, 1309} (no further terms, since numbers divisible by 4 cannot be sphenic numbers) least start of exactly n consecutive numbers that are Achilles numbers: {72, 5425069447, {105, 66, 406} least start of exactly n consecutive triangular numbers that are sphenic numbers (no further terms, since if 4 consecutive triangular numbers are all sphenic numbers, let them be n(n+1)/2, (n+1)(n+2)/2, (n+2)(n+3)/2, (n+3)(n+4)/2, then we have omega(n(n+1)) = omega((n+1)(n+2)) = omega((n+2)(n+3)) omega((n+3)(n+4) = 4, and none of n, n+1, n+2, n+3, n+4 is divisible by 8 (or at least one of n(n+1)/2, (n+1)(n+2)/2, (n+2)(n+3)/2, (n+3)(n+4)/2 will be divisible by 4 and cannot be sphenic number), thus omega(n) = omega(n+2) = omega(n+4), but n, n+2, n+4 cannot be all primes unless n=3, thus omega(n) must be >= 2, and if omega(n) = 2, then we have omega(n+1) = 2 (since omega(n(n+1)) = 4), similarly, omega(n+2) = omega(n+3) = omega(n+4) = 2, which is impossible since at least one of n, n+1, n+2, n+3, n+4 is divisible by 4, thus this number can only be 4, thus omega(n) must be 3, and omega(n+1) = omega(n+3) = 1, i.e. n+1 and n+3 are twin primes, thus neither n+1 nor n+3 is divisible by 3, thus n+2 is divisible by 3, and n+2 cannot be divisible by 12 since omega(n+2) = 3 and if n+2 is divisible by 12 then n+2 can only be 12, thus n and n+4 must be divisible by 4 (note that neither n+1 nor n+3 cannot be divisible by 4 since n+1 and n+3 are twin primes), and thus one of n and n+4 will be divisible by 8, which is a contradiction! The smallest prime factor of A005165(n), this record sequence is finite since for all n>=3612702, the smallest prime factor of A005165(n) are all 3612703, but we do not know the last record n and its record value (since this requires to factor near 22122513-digit numbers, the last record n and its record value will not be known in our lifetime! but we know that they really exist! this is equivalent to this problem: Find the smallest m such that gcd(A005165(n),m) > 1 for all n >= 2, such m must exist, but we cannot find the m in our lifetime) Last fiddled with by sweety439 on 2022-03-06 at 13:59

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2022-04-24, 20:29	#3
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 3682₁₀ Posts	Related page: What is special about this number? (Some numbers, like 43 and 94, breaks a law of small numbers, also some numbers, like 24 and 41, are the (proved) largest number satisfying a property.