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

0,8

Comments

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

Links

Table of n, a(n) for n=0..80.

Formula

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

Example

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

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 *)

Crossrefs

Cf. A025426, A317682 - A317684.

Sequence in context: A281530 A340260 A175190 * A257540 A333516 A228202

Adjacent sequences: A317682 A317683 A317684 * A317686 A317687 A317688

Keyword

nonn,easy

Author

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

Status

approved