A091824 Numbers p such that ((p-1)!*2^(p-1) + 1)/p is a prime.

1, 3, 37, 61, 10651

Offset

1,2

Comments

1 is not prime, but (1-1)!*2^(1-1) + 1 = 2 is a prime, so 1 is in the sequence.

If p is a prime and gcd(q,p)=1, then p divides (p-1)!*q^(p-1) + 1.

Links

Table of n, a(n) for n=1..5.

Ke Xiao, Dimensionless Physical Constant Mysteries, viXra:1205.0050, 2012. - From N. J. A. Sloane, Oct 15 2012

Example

((1-1)!*2^(1-1) + 1)/1 = 2 is a prime, so 1 is the first term in the sequence.
((2-1)!*2^(2-1) + 1)/2 = is not an integer, so 2 is not in the sequence.
((3-1)!*2^(3-1) + 1)/3 = 3 is a prime, so 3 is the second term in the sequence.
The next number p that yields a prime is 37:
((37-1)!*2^(37-1) +1)/37 = 690896939629347629014331483828706966091078572972973.

Prog

(PARI) for (i=3, 1100, if(isprime(((i-1)!*2^(i-1)+1)/i), print(i)));
(Magma) [1] cat [p: p in PrimesUpTo(10000) | IsPrime((Factorial(p-1)*2^(p-1) + 1) div p )]; // Vincenzo Librandi, Aug 21 2015

Crossrefs

Cf. A091825.

Sequence in context: A047691 A323829 A139842 * A056408 A056398 A092074

Adjacent sequences: A091821 A091822 A091823 * A091825 A091826 A091827

Keyword

hard,more,nonn

Author

Mohammed Bouayoun (mohammed.bouayoun(AT)sanef.com), Mar 09 2004

Extensions

a(5) from Michael S. Branicky, Apr 24 2023

Status

approved