A159266 Least positive integer m such that 2^n+3^m is prime.

1, 1, 1, 1, 2, 1, 1, 3, 2, 2, 4, 1, 3, 3, 1, 1, 8, 1, 4, 8, 8, 6, 2, 7, 27, 6, 13, 1, 10, 1, 6, 4, 8, 18, 9, 19, 2, 15, 29, 3, 3, 17, 10, 3, 11, 6, 2, 5, 20, 34, 4, 12, 10, 26, 1, 4, 2, 9, 29, 29, 10, 34, 13, 4, 8, 2, 1, 8, 10, 26, 50, 19, 12, 10, 8, 13, 27, 17, 9, 33, 4, 2, 17, 1, 7, 3, 5, 61, 26

Offset: 1

Author

M. F. Hasler, Apr 07 2009

Keywords

• nonn

Comments

In contrast to A123340 which allows m=0, a(0) does not exist for this sequence.

Examples

			a(1)=1 is the least m>0 such that 2^1+3^m (=5) is prime.
a(2)=1 is the least m>0 such that 2^2+3^m (=7) is prime.
a(5)=2 is the least m>0 such that 2^5+3^m (=41) is prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..2000
• M. F. Hasler, Primes of the form (x+1)^p-x^p, Apr 7, 2009.
• Maximilian Hasler, Mike Oakes, Mark Underwood, David Broadhurst and others, Primes of the form (x+1)^p-x^p, digest of 22 messages in primenumbers Yahoo group, Apr 5 - May 7, 2009. [Cached copy]

Crossrefs

Cf. A123340 (allows for m=0), A123359 (roles of 2 and 3 exchanged).

Programs

Maple

f:= proc(n) local t,m;
  t:= 2^n;
  for m from 1 do if isprime(t+3^m) then return m fi od
end proc:
map(f, [$1..100]); # Robert Israel, Sep 18 2018

Mathematica

a[n_] := Module[{m, t = 2^n}, For[m = 1, True, m++, If[PrimeQ[t + 3^m], Return[m]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 01 2023 *)

PARI

A159266(n,m=0)=until( is/*pseudo*/prime(2^n+3^m++),);m /* 2nd optional arg allows us to resume search after a given m and thus (when set to previous result) the list of all m yielding primes */

Formula

a(n) = min { m>0 | 2^n+3^m is prime } = A123340(n) whenever the latter is > 1.