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A059843 a(n) is the smallest prime p such that p-n is a nonzero square. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
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
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
These terms arise in A002496, A056899, A049423, A005473, A056905, A056909 as first or 2nd entries depending on offset.
Cf. A056896 (where p-n can be 0).
Sequence in context: A332211 A353075 A069587 * A092927 A071553 A021812
KEYWORD
nonn
AUTHOR
Labos Elemer, Feb 26 2001
STATUS
approved

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Last modified July 23 08:15 EDT 2024. Contains 374546 sequences. (Running on oeis4.)