A317682 Number of partitions of n into a prime and two distinct squares.

0, 0, 0, 1, 1, 0, 2, 2, 2, 1, 1, 2, 4, 1, 2, 3, 3, 2, 4, 2, 4, 3, 4, 4, 4, 1, 2, 6, 6, 3, 5, 3, 6, 5, 3, 2, 7, 3, 5, 7, 4, 4, 8, 5, 6, 5, 5, 7, 9, 3, 4, 6, 7, 6, 9, 5, 8, 9, 6, 4, 9, 3, 6, 11, 6, 5, 10, 7, 10, 8, 8, 8, 12, 5, 5, 8, 10, 9, 11, 6, 7

Offset: 0

Author

R. J. Mathar, Michel Marcus, Aug 04 2018

Keywords

• nonn
• easy

Comments

As in A025435, zero is a valid square here.

Examples

			a(12)=4 counts 12 = 11 + 0^2 + 1^2 = 3 + 0^2 + 3^2 = 7 + 1^2 + 2^2 = 2 + 1^2 + 3^2.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..20000

Crossrefs

Cf. A025435, A317683 - A317685.

Programs

Maple

A317682 := proc(n)
    a := 0 ;
    p := 2;
    while p < n do
        a := a+A025435(n-p);
        p := nextprime(p) ;
    end do:
    a ;
end proc:

Mathematica

A025435[n_] := Length[ PowersRepresentations[n, 2, 2]] - Boole[ IntegerQ[ Sqrt[2n]]];
a[n_] := Module[{s = 0, p}, For[p = 2, p <= n-1, p = NextPrime[p], s += A025435[n-p]]; s];
a /@ Range[0, 100] (* Jean-François Alcover, Apr 07 2020 *)

PARI

A317682(n,s=0)={forprime(p=2,n-1,s+=A025435(n-p));s} \\ M. F. Hasler, Aug 05 2018

Formula

a(n) = Sum_{primes p} A025435(n-p).