A091669 a(n) = (2^(n-1)/n!) * Product_{k=1..n-1} (2^k-1).

1, 1, 2, 7, 42, 434, 7812, 248031, 14055090, 1436430198, 267176016828, 91151551074486, 57425477176926180, 67196011936600334340, 146782968474309770332296, 601204690999713530559792879

Offset: 1

Author

Karol A. Penson, Jan 27 2004

Keywords

• nonn

Comments

Primes p such that 2^p-2 divides a(p) are A216838. - Amiram Eldar and Thomas Ordowski, Jan 16 2020

For odd n > 1, if a(n-1) divides a(n) and n does not divide a(n), then n is a prime (for which 2 is a primitive root, A001122). Composite numbers m such that a(m-1) divides a(m) are the pseudoprimes A001567 and A006935. Numbers n > 1 such that a(m) divides a(n) for all m < n are primes 2, 3, 5, 7, and 13. These are the primes p for which gpf(2^p-2) = p. - Thomas Ordowski, Jan 17 2020

If p is a prime with primitive root 2, A001122, then p | a(p-1) + 2^(p-2). Conjecture: (for n > 2), if n | a(n-1) + 2^(n-2), then n is a prime (A001122). Note that if p is an odd prime for which 2 is not a primitive root, A216838, then p | a(p-1). - Amiram Eldar and Thomas Ordowski, Jan 19 2020

Links

• Amiram Eldar, Table of n, a(n) for n = 1..86

Crossrefs

Cf. A000142, A001122, A001567, A005329, A006935, A159353, A216838, A225101.

Programs

Magma

[1] cat [2^(n-1)/Factorial(n)*&*[(2^k-1):k in [1..n-1]]:n in [2..16]]; // Marius A. Burtea, Jan 16 2020

Maple

seq( (2^(n-1)/n!)*mul(2^j-1, j=1..n-1), n=1..20); # G. C. Greubel, Feb 05 2020

Mathematica

Table[QFactorial[n-1, 2] 2^(n-1)/n!, {n, 20}]

PARI

a(n) = (2^(n-1)/n!) * prod(k=1, n-1, 2^k-1); \\ Michel Marcus, Jan 16 2020

Sage

from sage.combinat.q_analogues import q_factorial
[2^(n-1)*q_factorial(n-1, 2)/factorial(n) for n in (1..20)] # G. C. Greubel, Feb 05 2020

Formula

a(n) = 2^(n-1)*A005329(n-1)/n!.

a(n) = Product_{k=2..n} (2^k-2)/k = Product_{k=2..n} A225101(k)/A159353(k). - Thomas Ordowski, Jan 16 2020

Extensions

Corrected and edited by Thomas Ordowski, Jan 16 2020