A189373 Perfect numbers k such that k+1 is prime.

6, 28, 33550336, 137438691328

Offset: 1

Author

Luis H. Gallardo, Apr 23 2011

Keywords

• nonn
• bref
• hard

Comments

Joerg Arndt checked that up to exponent p=110503 of the corresponding Mersenne prime 2^p - 1 the number k=2^(p-1)*(2^p-1)+1 is not pseudoprime.

The listed perfect numbers have exponents p in 2, 3, 13, 19.

Examples

			We have a(3) = 33550336 since 33550337 is prime and there is no other such perfect number less than a(3) and that exceeds a(2) = 28.
		

Links

• MathOverflow, Even perfect numbers n with n+1 prime.

Crossrefs

Cf. A000396, A061644, A099057.

Programs

PARI

{e=[2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787];} /* exponents of Mersenne primes */
for(n=1,#e,p=(2^e[n]-1)*(2^(e[n]-1));if(ispseudoprime(p+1),print1(p,", ")));

Formula

a(n) = A061644(n) - 1. - Amiram Eldar, May 06 2024