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A350381
Composite numbers k such that the multiplicative order of 2 modulo lpf(2^k-1) is k, where lpf = least prime factor.
0
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
OFFSET
1,1
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.
EXAMPLE
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.
PROG
(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.
CROSSREFS
Cf. A292559, A322568, A049479 (lpf(2^n-1)).
Sequence in context: A124979 A292559 A322568 * A018820 A231632 A202004
KEYWORD
nonn,hard,more
AUTHOR
Jianing Song, Dec 28 2021
STATUS
approved