A173854 Smallest positive integer k such that 2^n + k^2 is a prime number.

1, 1, 1, 3, 1, 3, 3, 3, 1, 3, 3, 9, 9, 9, 7, 15, 1, 15, 3, 9, 5, 21, 5, 3, 11, 57, 7, 21, 9, 33, 3, 27, 9, 15, 5, 39, 25, 3, 35, 57, 25, 9, 15, 33, 39, 99, 27, 3, 25, 63, 67, 9, 105, 51, 145, 33, 9, 3, 15, 57, 15, 243, 13, 111, 9, 15, 3, 81, 71, 21, 5, 21, 19, 33, 57, 81, 141, 51, 17, 33, 125

Offset: 0

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Feb 26 2010

Keywords

• nonn

Comments

The list of associated primes 2^n + k^2 is 2, 3, 5, 17, 17, 41, 73, 137, 257, 521, 1033, ...

All terms are odd. - Harvey P. Dale, Dec 19 2014

Examples

			2^0 + 1^2 = 2 = A000040(1) => a(0) = k = 1
2^1 + 1^2 = 3 = A000040(2) => a(1) = k = 1
2^2 + 1^2 = 5 = A000040(3) => a(2) = k = 1
2^3 + 3^2 = 17 = A000040(7) => a(3) = k = 3
2^61 + 243^2 = A000040(tbd) => a(61) = k = 243.
		

References

• Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005
• Louis J. Mordell: Diophantine equations, Academic Press Inc., 1969
• Wolfgang M. Schmidt, Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics vol. 785, Springer-Verlag, 2000

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A013597, A014210.

Programs

Maple

A173854 := proc(n) local twon,k ; twon := 2^n ; for k from 1 do if isprime(twon+k^2) then return k ; end if; end do ; end proc:
seq(A173854(n),n=0..90) ; # R. J. Mathar, Mar 05 2010

Mathematica

spi[n_]:=Module[{t=2^n,k=1},While[!PrimeQ[t+k^2],k=k+2];k]; Array[spi,90,0] (* Harvey P. Dale, Dec 19 2014 *)

Extensions

Extended by R. J. Mathar, Mar 05 2010