A013597 a(n) = nextprime(2^n) - 2^n.

1, 1, 1, 3, 1, 5, 3, 3, 1, 9, 7, 5, 3, 17, 27, 3, 1, 29, 3, 21, 7, 17, 15, 9, 43, 35, 15, 29, 3, 11, 3, 11, 15, 17, 25, 53, 31, 9, 7, 23, 15, 27, 15, 29, 7, 59, 15, 5, 21, 69, 55, 21, 21, 5, 159, 3, 81, 9, 69, 131, 33, 15, 135, 29, 13, 131, 9, 3, 33, 29, 25, 11, 15, 29

Offset: 0

Author

James Kilfiger (mapdn(AT)csv.warwick.ac.uk)

Keywords

• nonn

Comments

A013597 and A092131 use different definitions of "nextprime(2)", namely A151800 vs A007918: A013597 assumes nextprime(2) = 3 = A151800(2), whereas A092131 assumes nextprime(2) = 2 = A007918(n). [Edited by M. F. Hasler, Sep 09 2015]

If (for n>0) a(n)=1, then n is a power of 2 and 2^n+1 is a Fermat prime. n=1,2,4,8,16 are probably the only indices with this property. - Franz Vrabec, Sep 27 2005

Conjecture: there are no Sierpiński numbers in the sequence. See A076336. - Thomas Ordowski, Aug 13 2017

Links

• T. D. Noe, Table of n, a(n) for n = 0..5000
• V. Danilov, Table for large n

Crossrefs

Cf. A014210, A092131, A007918, A151800.

Programs

Maple

A013597 := proc(n)
    nextprime(2^n)-2^n ;
end proc:
seq(A013597(n),n=0..40) ;

Mathematica

Table[NextPrime[#] - # &[2^n], {n, 0, 73}] (* Michael De Vlieger, Aug 15 2017 *)

PARI

a(n) = nextprime(2^n+1) - 2^n; \\ Michel Marcus, Nov 06 2015

Python

from sympy import nextprime
def A013597(n): return nextprime(m:=1<Chai Wah Wu, Dec 02 2024

Formula

a(n) = A151800(2^n) - 2^n = A013632(2^n). - R. J. Mathar, Nov 28 2016

Conjecture: a(n) < n^2/2 for n > 1. - Thomas Ordowski, Aug 13 2017