A345331 Odd numbers k > 1 such that m^(2^v(k-1)+1) == -m (mod k) has more than one solution modulo k, where v(k) = A007814(k) is the 2-adic valuation of k.

15, 35, 39, 51, 55, 75, 85, 87, 91, 95, 111, 115, 119, 123, 135, 143, 153, 155, 159, 175, 183, 187, 195, 203, 205, 215, 219, 221, 235, 247, 255, 259, 267, 275, 287, 291, 295, 299, 303, 315, 319, 323, 327, 335, 339, 351, 355, 357, 365, 371, 375, 391, 395, 403

Offset

1,1

Comments

Note that for even k, m == -1 (mod k) is a solution.

All terms are composite.

Odd composite k is a term if and only if v(p-1) > v(k-1) for some prime factors p of k. See A345330 for a proof.

This sequence and the Carmichael numbers (A002997) are disjoint: if k is a Carmichael number, then p-1 | k-1 for all prime factors p.

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Example

51 is a term since 51 = 3 * 17 and v(17-1) = 4 > v(51-1) = 1. Also, m^(2^v(51-1)+1) == -m (mod 51) has three solutions: m == 0, 21, 30 (mod 51).

Prog

(PARI) isA345331(n) = if(!isprime(n) && n>1 && n%2, my(f=factor(n), w=omega(n)); for(i=1, w, if(valuation(f[i, 1]-1, 2) > valuation(n-1, 2), return(1))); 0, 0)

Crossrefs

Cf. A007814, A002997.

Complement of A345330 with respect to A071904.

Sequence in context: A329589 A356794 A154988 * A360110 A359163 A327934

Adjacent sequences: A345328 A345329 A345330 * A345332 A345333 A345334

Keyword

nonn

Author

Jianing Song, Jun 14 2021

Status

approved