A322568 Integers k such that the least prime factor of 2^k - 1 is not in A122094.

169, 221, 323, 611, 779, 793, 923, 1121, 1159, 1271, 1273, 1349, 1513, 1717, 1829, 1919, 2033, 2077, 2197, 2201, 2413, 2533, 2603, 2759, 2873, 2951, 3097, 3131, 3173, 3193, 3211, 3281, 3379, 3599, 3721, 3757, 3791, 3937, 3953, 4043, 4199, 4223, 4309, 4331

Offset: 1

Author

Jeppe Stig Nielsen, Aug 29 2019

Keywords

• nonn

Comments

Clearly, the terms are odd and composite (A071904).

The first term which is itself of form 2^j - 1 is 34359738367 = 2^35 - 1. The least prime factor of 2^34359738367 - 1 is 136463, and the multiplicative order of 2 modulo 136463 is 2201 = 31*71. In A309130, it is asked if a member of A322568 can be of form 2^p - 1 with p prime.

Examples

			169 is included because the least prime factor of 2^169-1 is 4057, and the multiplicative order of 2 modulo 4057 is 169 which is not prime. The divisor 4057 is less than the "algebraic" divisor 2^13-1 = 8192 (Mersenne prime).
4199 (= 13*17*19) is included because the least prime factor of 2^4199-1 is 647, and the multiplicative order of 2 modulo 647 is 323 (= 17*19) which is not prime. The divisor 647 is less than the smallest "algebraic" divisor which is 2^13-1 = 8192 (Mersenne prime).
289 is NOT included; its least prime factor is 2^17 - 1.
1073 (= 29*37) is NOT included; its least prime factor is 223, but 223 is a divisor of one of the "algebraic" factors, namely 223 is a divisor of composite Mersenne number 2^37 - 1.
		

Crossrefs

Cf. A000225, A020639, A122094, A309130.

Programs

PARI

for(k=2,+oo,isprime(k)&&next();forprime(p=3,,if(Mod(2,p)^k-1==0,!isprime(znorder(Mod(2,p)))&&print1(k,", ");next(2))))