cp's OEIS Frontend

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

a(2), a(3), a(5) and a(7) are prime; a(11) is not.

Let S be a set of n elements. First we perform a set partition on S. Let SU be the set of all nonempty subsets of S. As is well known, 2^n - 1 is the number of nonempty subsets of a set with n elements (see A000225). That is, 2^n - 1 is the number of elements |SU| of SU. In the second step, we select k elements from SU. We want to know how many different selections are possible. Let W be the resulting set of selections formed from SU. Then the number of elements |W| of W is |W| = Sum_{k=1..2^n-1} binomial(2^n-1,k) = 2^(2^n-1) - 1 = A077585. Example: |W(n)| = a(n=2) = 7, because W = {[[1, 2]], [[1]], [[1, 2], [1]], [[1, 2], [2], [1]], [[1, 2], [2]], [[2], [1]], [[2]]}. - Thomas Wieder, Nov 08 2007

These are so-called double Mersenne numbers, sometimes denoted MM(n) where M = A000225. MM(n) is prime iff M(n) = 2^n-1 is a Mersenne prime exponent (A000043), which isn't possible unless n itself is also in A000043. Primes of this form are called double Mersenne primes MM(p). For all Mersenne exponents between 7 and 61, factors of MM(p) are known. MM(61) is far too large for any currently known primality test, but a distributed search for factors of this and other MM(p) is ongoing, see the doublemersenne.org web site. - M. F. Hasler, Mar 05 2020

This is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Dec 05 2022

Same as exponents of double Mersenne primes. Only four terms are known.

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)

Only four terms are known.

The first four Mersenne primes (p=2^q-1 in A000668) are double Mersenne primes, i.e., in A103901. The next four yield a composite M(p) and therefore are in this sequence. The next larger Mersenne prime p = A000668(9) has already 19 digits and is much too large to enable us, as of today, to test the primality of 2^p-1 (which would require over 10^8 gigabytes just to be stored in binary). This explains that only 4 terms are known of this sequence and of A103901; for all the 30+ remaining members of A000668 it is not known whether they belong to A103901 or to this sequence A103902. - M. F. Hasler, Jan 21 2015

The terms of this sequence are sometimes called "Double Mersenne numbers" (cf. A263686).

Agrees with A077586 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(5) is too large to include in data section (see A276641).

a(n) = A263686(n) iff a(n) is prime, which is the case iff A000668(n) is in A103901.

Agrees with A263686 at least in the first four terms. - Omar E. Pol, Oct 24 2016