A108714 a(n) is the minimal value of k, such that n^2+k^2 or (n^2+k^2)/2 are primes.

1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 7, 2, 1, 1, 1, 2, 5, 1, 1, 4, 5, 3, 1, 1, 1, 2, 5, 1, 11, 4, 3, 2, 5, 1, 1, 2, 3, 1, 1, 4, 5, 3, 9, 1, 5, 2, 13, 1, 7, 1, 3, 3, 1, 4, 1, 2, 3, 1, 13, 1, 3, 5, 9, 1, 1, 2, 5, 1, 3, 1, 5, 2, 1, 4, 5, 3, 7, 1, 7, 4, 3, 3, 1, 1, 9, 2

Offset

1,7

Comments

I am attempting to complete a proof that for every natural number n, there is at least one prime of the form n^2+k^2 or (n^2+k^2)/2 with 1<=k<=n.

Links

Table of n, a(n) for n=1..87.

Example

a(3) = 1 because (3^2+1)/2 = 5 (prime);
a(7) = 2 ------> 7^2+2^2 = 53 (prime);
a(12) = 7 -----> 12^2+7^2 = 193 (prime);
a(23) = 3 -----> (23^2+3^2)/2 = 269 (prime);
a(48) = 13 ----> 48^2+13^2 = 2473 (prime);
...

Mathematica

a[n_]:=Module[{k=1}, While[!PrimeQ[n^2+k^2]&&!PrimeQ[(n^2+k^2)/2], k++]; k]; Array[a, 87] (* Stefano Spezia, Sep 09 2025 *)

Crossrefs

Cf. A070216.

Sequence in context: A366292 A205104 A215561 * A135508 A030413 A139434

Adjacent sequences: A108711 A108712 A108713 * A108715 A108716 A108717

Keyword

nonn

Author

Robin Garcia, Jun 20 2005

Extensions

a(54)-a(87) from Stefano Spezia, Sep 09 2025

Status

approved