A090748 Numbers k such that 2^(k+1) - 1 is prime.

1, 2, 4, 6, 12, 16, 18, 30, 60, 88, 106, 126, 520, 606, 1278, 2202, 2280, 3216, 4252, 4422, 9688, 9940, 11212, 19936, 21700, 23208, 44496, 86242, 110502, 132048, 216090, 756838, 859432, 1257786, 1398268, 2976220, 3021376, 6972592, 13466916, 20996010, 24036582, 25964950, 30402456, 32582656

Offset: 1

Author

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Feb 03 2004

Keywords

• nonn

Comments

Perfect numbers A000396(n) = 2^A133033(n) - 2^a(n), assuming there are no odd perfect numbers. - Omar E. Pol, Feb 24 2008

Number of proper divisors of n-th even perfect number that are multiples of n-th Mersenne prime A000668(n). - Omar E. Pol, Feb 28 2008

Base 2 logarithm of n-th even superperfect number A061652(n). Also base 2 logarithm of n-th superperfect number A019279(n), assuming there are no odd superperfect numbers. - Omar E. Pol, Apr 11 2008

Number of 0's in binary expansion of n-th even perfect number (see A135650). - Omar E. Pol, May 04 2008

Examples

			1 is in the sequence because 2^2 - 1 = 3 is prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..48 (terms 1..47 from Ivan Panchenko)
• Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Crossrefs

a(n) = A000043(n) - 1. A000043 is the main entry for this sequence.

Cf. A000396, A133033, A000668, A019279, A061652, A135650, A051027.

Programs

Magma

[n: n in [1..5*10^3] |IsPrime(2^(n+1)-1)]; // Vincenzo Librandi, Jul 28 2016

Mathematica

Select[Range[0, 10^4], PrimeQ[2^(# + 1) - 1] &] (* Vincenzo Librandi, Jul 28 2016 *)

PARI

is(n)=ispseudoprime(2^(n+1)-1) \\ Charles R Greathouse IV, Aug 21 2016

Formula

a(n) = A000043(n) - 1.

2^(a(n) + 1) = A051027(2^a(n)). - Juri-Stepan Gerasimov, Aug 21 2016 [corrected by Jerzy R Borysowicz, Feb 26 2025]

Extensions

Edited, corrected and extended by Robert G. Wilson v, Feb 09 2004

a(39) from Omar E. Pol, Jan 23 2009

a(40)-a(44) from Ivan Panchenko, Apr 11 2018