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

Author

Jianing Song, Jun 14 2021

Keywords

• nonn

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.

Examples

			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).
		

Links

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

Crossrefs

Cf. A007814, A002997.

Complement of A345330 with respect to A071904.

Programs

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)