A059843 a(n) is the smallest prime p such that p-n is a nonzero square.

2, 3, 7, 5, 41, 7, 11, 17, 13, 11, 47, 13, 17, 23, 19, 17, 53, 19, 23, 29, 37, 23, 59, 73, 29, 107, 31, 29, 173, 31, 47, 41, 37, 43, 71, 37, 41, 47, 43, 41, 617, 43, 47, 53, 61, 47, 83, 73, 53, 59, 67, 53, 89, 79, 59, 137, 61, 59, 383, 61, 97, 71, 67, 73, 101, 67, 71, 149, 73

Offset: 1

Author

Labos Elemer, Feb 26 2001

Keywords

• nonn

Examples

			For n = 17, let P = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,...} be the set of primes, then P - 17 = {-15,...,-4,0,2,6,12,14,20,24,26,30,36,...}. The first positive square in P - 17 is 36 with p = 53, so a(17) = 53. The square arising here is usually 1.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

These terms arise in A002496, A056899, A049423, A005473, A056905, A056909 as first or 2nd entries depending on offset.

Cf. A002496, A056899, A049423, A005473, A056905, A056909.

Cf. A056896 (where p-n can be 0).

Programs

Maple

SearchLimit := 100;
for n from 1 to 400 do
k := 0: c := true:
while(c and k < SearchLimit) do
    k := k + 1:
    c := not isprime(k^2+n):
end do:
if k = SearchLimit then error("Search limit reached!") fi;
a[n] := k^2 + n end do: seq(a[j], j=1..400);
# Edited and SearchLimit introduced by Peter Luschny, Feb 05 2019

Mathematica

spsq[n_]:=Module[{p=NextPrime[n]},While[!IntegerQ[Sqrt[p-n]],p= NextPrime[ p]];p]; Array[spsq,70] (* Harvey P. Dale, Nov 10 2017 *)

PARI

for(n=1, 100, for(k=1, 100, if(isprime(k^2+n), print1(k^2+n, ", "); break()))) \\ Jianing Song, Feb 04 2019

PARI

a(n) = forprime(p=n,, if ((p-n) && issquare(p-n), return (p))); \\ Michel Marcus, Feb 05 2019

Formula

a(n) = min{p : p - n = x^2 for some x > 0, p is prime}.

Does a(n) exist for all n? - Jianing Song, Feb 04 2019