A309130 -id:A309130 - cp's OEIS frontend

A263686 Smallest prime factor of double Mersenne numbers.

7, 127, 2147483647, 170141183460469231731687303715884105727, 338193759479, 231733529, 62914441, 295257526626031

Offset: 1

Arkadiusz Wesolowski, Oct 23 2015

A double Mersenne number is a Mersenne number of the form 2^(2^p - 1) - 1, where p is a Mersenne exponent (A000043).
From M. F. Hasler, Feb 28 2025: (Start)
The prime factors of Mersenne numbers 2^q - 1 must be of the form 2*q*k + 1.
The four smallest double Mersenne numbers (p = 2, 3, 5, 7 => q = 3, 7, 31, 127) are prime, so their smallest prime factor is equal to themselves, a(n) = M(q). This is equivalent to k = (2^(q-1)-1)/q, which is almost as large as M(q) itself: k = 1, 9 and 34636833 for the first three terms, and for q = 127, k has just three digits less than M(q) = a(4) itself. The prime p = 11 is not a Mersenne exponent.
The fifth term, a(5) = 2*(2^13-1)*k + 1 with k = 20644229 (which is prime) is the first proper divisor of the respective M(q), as are the next three, corresponding to p = 17, 19 and 31.
For p = 61, M(q) has 694127911065419642 digits, and so far no factor is known, but it is known that it has no factor less than 10^36. (End)

Double Mersennes Prime Search, History
Wikipedia, Double Mersenne number

Cf. A000043, A000668, A001348, A020639, A049479, A077586, A122094. Subsequence of A016047. Subsequence of A309130.

forprime(p=2,,q=2^p-1; !ispseudoprime(q) && next(); if(ispseudoprime(2^q-1), print1(2^q-1,", ");next()); forstep(r=2*q+1,+oo,2*q, !ispseudoprime(r) && next(); if(Mod(2,r)^q-1 == 0, print1(r,", ");next(2)))) \\ Jeppe Stig Nielsen, Aug 28 2019

a(n) = spf(MM(A000043(n))) = A049479(A000668(n)), where spf = A020639 is the smallest prime factor, A049479 = spf o M, M(p) = 2^p-1 = A000225(p), MM = M o M = A077585, A000668(n) = M(A000043(n)), A000043 are the Mersenne prime exponents. - M. F. Hasler, Mar 01 2025

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

Original entry on oeis.org

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

Jeppe Stig Nielsen, Aug 29 2019

nonn

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.

			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.

Cf. A000225, A020639, A122094, A309130.

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))))

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A263686 Smallest prime factor of double Mersenne numbers.

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