A108574 Range of A000790 (primary pretenders).

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 38, 39, 46, 49, 51, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 106, 111, 118, 121, 122, 123, 129, 133, 134, 141, 142, 145, 146, 158, 159, 166, 169, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 217, 218, 219, 226, 237, 249, 254, 259, 262, 265, 267, 274, 278, 289, 291, 298, 301, 302, 303, 305, 309, 314, 321, 326, 327, 334, 339, 341, 346, 358, 361, 362, 365, 381, 382, 386, 393, 394, 398, 411, 417, 422, 427, 445, 446, 447, 451, 453, 454, 458, 466, 469, 471, 478, 481, 482, 485, 489, 501, 502, 505, 511, 514, 519, 526, 529, 537, 538, 542, 543, 545, 553, 554, 561

Offset: 1

Author

David W. Wilson, Jun 10 2005

Keywords

• fini
• full
• nonn

Comments

All terms except for the last term, 561, are semiprimes (A001358). Semiprimes up to 559 that are not here: 35, 55, 77, 95, 115, 119, 143, 155, 161, 187, 203, 209, 215, 221, 235, 247, 253, 287, 295, 299, 319, 323, 329, 335, 355, 371, 377, 391, 395, 403, 407, 413, 415, 437, 473, 493, 497, 515, 517, 527, 533, 535, 551, 559. - Zak Seidov, Jan 08 2015

The LCM of all terms is 23# * 277# (where # denotes the primorial function A034386), the period of A000790, and therefore also of the related sequence b(n) = gcd(A000790(n), n). - M. F. Hasler, Feb 16 2018

Range of A295997. - Thomas Ordowski, Feb 27 2018

These numbers k < 561 are semiprimes k = pq such that p-1 | q-1, where primes p <= q. Equivalent condition is p-1 | k-1. - Thomas Ordowski, Aug 18 2018

This shows that all even semiprimes < 561 are in this sequence. The odd semiprimes not in this sequence are the semiprimes (equivalently: all terms but 275, 455, 475, 539) less than 561 in A267999 (which equals A121707 up to 695). - M. F. Hasler, Nov 09 2018

Links

• J. H. Conway, R. K. Guy, W. A. Schneeberger and N. J. A. Sloane, The Primary Pretenders, Acta Arith. 78 (1997), 307-313.
• J. H. Conway, R. K. Guy, W. A. Schneeberger and N. J. A. Sloane, The Primary Pretenders, arXiv:math/0207180 [math.NT], 2002.

Crossrefs

Cf. A000790, A001358, A295997.

Programs

Mathematica

pp[n_] := For[c = 4, True, c = If[PrimeQ[c+1], c+2, c+1], If[PowerMod[n, c, c] == Mod[n, c], Return[c]]];seq[n_] := seq[n] = Table[pp[k], {k, 0, 2^n}] // Union; seq[10]; seq[n = 11]; While[ Print["n = ", n, " more terms: ", Complement[seq[n], seq[n-1]]]; seq[n] != seq[n-1], n++]; A108574 = seq[n] (* Jean-François Alcover, Oct 18 2013 *)

PARI

my(A=List(561)); forprime(q=2,561\2, forprime(p=2,min(q,561\q), (q-1)%(p-1)|| listput(A, p*q))); A108574=Set(A) \\ M. F. Hasler, Nov 09 2018