A260870 Least k>0 such that k^2 + (2n+1-k)^2 is prime, or 0 if no such k exists.

1, 1, 1, 2, 1, 3, 1, 1, 2, 1, 3, 1, 1, 2, 5, 4, 2, 1, 2, 1, 3, 4, 2, 2, 5, 4, 1, 1, 2, 3, 5, 3, 1, 2, 6, 3, 1, 5, 4, 5, 4, 1, 2, 2, 1, 4, 1, 2, 2, 3, 3, 2, 5, 7, 1, 3, 3, 1, 2, 1, 4, 1, 1, 4, 1, 4, 1, 2, 2, 5, 3, 3, 1, 2, 1, 5, 4, 1, 5, 1, 3, 2, 10, 2, 1, 3, 6, 1, 2, 1, 4, 1, 5, 10, 3, 3, 2, 10, 7

Offset

1,4

Comments

It appears that any odd number N = 2n+1 > 1 (and also N = 2, hence all primes, see A260869) can be written as the sum of two positive integers such that the sum of their squares is prime. For an even number > 2 this is obviously not possible since k and 2n-k have the same parity and therefore the sum of their squares is even.

The record values 1, 2, 3, 5, 6, 7, 10, 13, 16, 29, 30, 37, 40, 41, 49, 55, 64, 67, 68, 74, 85, 88, 106, 128, ... occur for indices n (half of the odd numbers 2n+1) 1, 4, 6, 15, 35, 54, 83, 121, 172, 281, 936, 1093, 1150, 1240, 3121, 4126, 5116, 6793, 11935, 12556, 18238, 32710, 33343, 57256, ...

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Mathematica

lk[n_]:=Module[{k=1}, While[!PrimeQ[k^2+(2n+1-k)^2], k++]; k]; Array[lk, 100] (* Harvey P. Dale, May 31 2017 *)

Prog

(PARI) A260870(n)=for(k=1, (n=2*n+1)\2, isprime(k^2+(n-k)^2)&&return(k))

Crossrefs

Sequence in context: A226006 A210943 A260869 * A282497 A087157 A351094

Adjacent sequences: A260867 A260868 A260869 * A260871 A260872 A260873

Keyword

nonn

Author

M. F. Hasler, Aug 09 2015

Status

approved