A322568 -id:A322568 - cp's OEIS frontend

A309130 Smallest prime factor of A077586(n).

7, 127, 2147483647, 170141183460469231731687303715884105727, 47, 338193759479, 231733529, 62914441, 2351, 1399, 295257526626031, 18287, 106937, 863, 4703, 138863, 22590223644617

Offset: 1

Richard N. Smith, Jul 13 2019

A263686 is a subsequence.
Agrees with A263686 in the first four terms, but then the two sequences differ for the first time at n = 5, because prime(5) = 11 is not in A000043.
a(18) = A263686(9) is greater than 1.56*10^17*(2^61-1), see link.
a(n) = A077586(n) iff A077586(n) is prime, A077586(n) is prime for 1 <= n <= 4, but composite for 5 <= n <= 17. The status of A077586(18) = 2^(2^61-1)-1 is unknown. It is conjectured that A077586(n) is composite for all n >= 5.
a(20) = 456959, a(21) = 18384329, a(22) = 198839, a(23) = 2349023, a(24) = A263686(10) is greater than 1.25*10^16*(2^89-1).
Conjecture: All terms are in A122094 (all terms in A263686 are in A122094).
For examples related to that conjecture, see A322568. - Jeppe Stig Nielsen, Aug 29 2019
a(30) = 46559, a(32) = 23671, a(36) = 7151489, a(39) = 4698047, a(41) = 719, a(43) = 1440847, a(45) = 179689, a(47) = 11759383, a(48) = 23602441, a(50) = 9024439, a(51) = 28875361, a(52) = 6301423, a(54) = 2493983, a(56) = 33518137, a(59) = 6727783, a(66) = 95111, a(72) = 1439, a(73) = 99833, a(78) = 38119, a(81) = 26849, a(83) = 8258911, a(86) = 16173559, a(89) = 625343, a(93) = 9743. - Chai Wah Wu, Oct 16 2019

Double Mersennes Prime Search, History.
Eric Weisstein's World of Mathematics, Double Mersenne Number.

Cf. A077586, A263686, A001348, A049479.

A309130(n)=A020639(2^(2^prime(n)-1)-1) \\ For efficiency, use addprimes([large terms of this sequence]). - M. F. Hasler, Mar 01 2025

a(n) = A020639(A077586(n)).

a(n) = A049479(A001348(n)). - M. F. Hasler, Mar 01 2025

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

Original entry on oeis.org

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

Jianing Song, Dec 28 2021

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.

			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.

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

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.

More terms from Jinyuan Wang, Jan 22 2025

cp's OEIS Frontend

A309130 Smallest prime factor of A077586(n).

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