A317685 Number of partitions of n into a prime and two positive squares.

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

Offset: 0

Author

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

Keywords

• nonn
• look
• easy

Comments

As in A025426, the two squares do not need to be distinct.

Examples

			a(7) = 2 counts 7 = 5 + 1^2 + 1^2 = 2 + 1^2 + 2^2.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A025426, A317682 - A317684.

Programs

Maple

A317685 := proc(n)
    a := 0 ;
    p := 2;
    while p <= n do
        a := a+A025426(n-p);
        p := nextprime(p) ;
    end do:
    a ;
end proc:

Mathematica

p2sQ[{a_,b_,c_}]:=PrimeQ[a]&&AllTrue[{Sqrt[b],Sqrt[c]},IntegerQ]||PrimeQ[b] && AllTrue[{Sqrt[c],Sqrt[a]},IntegerQ]||PrimeQ[c]&&AllTrue[{Sqrt[b],Sqrt[a]},IntegerQ]; Table[Count[IntegerPartitions[n,{3}],?(p2sQ[#]&)],{n,0,80}] (* _Harvey P. Dale, Mar 10 2023 *)

Formula

a(n) = Sum_{primes p} A025426(n-p).