cp's OEIS Frontend

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

%I A350381 #17 Jan 23 2025 00:29:37
%S A350381 169,221,323,611,779,793,923,1121,1159,1271,1273,1349,1513,1717,1829,
%T A350381 1919,2033,2077,2201,2413,2533,2603,2759,2951,3097,3131,3173,3193,
%U A350381 3281,3379,3599,3721,3791,3937,3953,4043,4223,4309,4331,4607,4619,4867,4883,4981,5111
%N A350381 Composite numbers k such that the multiplicative order of 2 modulo lpf(2^k-1) is k, where lpf = least prime factor.
%C A350381 Obviously, if p is a prime, then the multiplicative order of 2 modulo lpf(2^p-1) is p.
%C A350381 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.
%C A350381 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.
%C A350381 Every term is coprime to 2, 3, 5, 7, 11 and 23.
%e A350381 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.
%e A350381 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.
%e A350381 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.
%o A350381 (PARI) b(n) = forprime(p=3, oo, if(n % znorder(Mod(2,p))==0, return(p)))
%o A350381 isA350381(n) = !isprime(n) && (n>1) && znorder(Mod(2,b(n)))==n \\ Warning: this program can only give the first 7 terms.
%Y A350381 Cf. A049479 (lpf(2^n-1)), A292559, A322568.
%K A350381 nonn,hard
%O A350381 1,1
%A A350381 _Jianing Song_, Dec 28 2021
%E A350381 More terms from _Jinyuan Wang_, Jan 22 2025