A350381 Composite numbers k such that the multiplicative order of 2 modulo lpf(2^k-1) is k, where lpf = least prime factor.

169, 221, 323, 611, 779, 793, 923, 1121, 1159, 1271, 1273, 1349, 1513, 1717, 1829, 1919, 2033, 2077, 2201, 2413, 2533, 2603, 2759, 2951, 3097, 3131, 3173, 3193, 3281, 3379, 3599, 3721, 3791, 3937, 3953, 4043, 4223, 4309, 4331, 4607, 4619, 4867, 4883, 4981, 5111

Offset: 1

Author

Jianing Song, Dec 28 2021

Keywords

• nonn
• hard

Comments

Obviously, if p is a prime, then the multiplicative order of 2 modulo lpf(2^p-1) is p.

It is easy to see that this is a subsequence of A292559 and A322568, so this sequence is included in the intersection of those two sequences. The inclusion is proper. 68231 is in A292559 and A322568 but not in this sequence: lpf(2^68231-1) = 136463 = 2*68231 + 1, the multiplicative order of 2 modulo 136463 is 2201 = 31 * 71 < 68231.

A semiprime in A322568 is in this sequence by definition. 20519, 48263, 63023, 138263, 216239, 341651, 421259, 480323 are examples of terms that are not semiprimes.

Every term is coprime to 2, 3, 5, 7, 11 and 23.

Examples

			169 is a term since the least prime factor of 2^169 - 1 is 4057, and the multiplicative order of 2 modulo 4057 is 169.
323 is a term since the least prime factor of 2^323 - 1 is 647, and the multiplicative order of 2 modulo 647 is 323.
1343 is not a term since the least prime factor of 2^1343 - 1 is 2687, and the multiplicative order of 2 modulo 2687 is 79 < 1343.
		

Crossrefs

Cf. A049479 (lpf(2^n-1)), A292559, A322568.

Programs

PARI

b(n) = forprime(p=3, oo, if(n % znorder(Mod(2,p))==0, return(p)))
isA350381(n) = !isprime(n) && (n>1) && znorder(Mod(2,b(n)))==n \\ Warning: this program can only give the first 7 terms.

Extensions

More terms from Jinyuan Wang, Jan 22 2025