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

0,7

Comments

As in A025435, zero is a valid square here.

Links

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

Formula

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

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A025435, A317683 - A317685.

Sequence in context: A356997 A171412 A248213 * A216651 A071338 A078826

Adjacent sequences: A317679 A317680 A317681 * A317683 A317684 A317685

Keyword

nonn,easy

Author

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

Status

approved