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

0,4

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

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

Example

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.

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 *)

Crossrefs

Cf. A013597, A014210.

Sequence in context: A132680 A105595 A072219 * A059789 A275367 A023136

Adjacent sequences: A173851 A173852 A173853 * A173855 A173856 A173857

Keyword

nonn

Author

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

Extensions

Extended by R. J. Mathar, Mar 05 2010

Status

approved