A394741 a(n) = number of triples (x, y, z) such that x^2 + y*z = n, where x, y, z are primes, not necessarily distinct, and y <= z.

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 2, 1, 1, 0, 0, 2, 2, 0, 0, 0, 1, 1, 1, 1, 0, 0, 2, 2, 2, 0, 0, 2, 2, 0, 1, 1, 2, 1, 0, 2, 2, 1, 0, 1, 2, 1, 0, 2, 1, 0, 2, 0, 3, 0, 0, 3, 3, 2, 1, 1, 2, 3, 0, 2, 1, 0, 1, 1, 3, 0, 1, 3, 1, 1, 0, 2, 0, 1, 1, 2, 3, 1, 0, 2, 2, 1, 1, 2, 2, 0, 0, 2, 3, 1, 1

Offset

0,14

Comments

Also, number of primes p such that n - p^2 is a semiprime. - M. F. Hasler, Apr 11 2026

Links

Table of n, a(n) for n=0..97.

Example

a(8) = 1 counts the triple (2, 2, 2): 8 = 2^2 + 2*2.
a(10) = 1 counts the triple (2, 2, 3): 10 = 2^2 + 2*3.
a(13) = 2 counts the triples (2, 3, 3) and (3, 2, 2): 13 = 2^2 + 3*3 = 3^2 + 2*2.

Mathematica

t[n_, c_] := Module[{r}, r = Flatten[Table[If[n - x^2 <= 0, {},
       Map[({x, #, Quotient[n - x^2, #]} &),
       Select[Divisors[n - x^2], Divisible[n - x^2, #] &]]], {x, 1,
       Floor[Sqrt[n - 1]]}], 1]; Select[r, Apply[c, #] &]];
c = (PrimeQ[#1] && PrimeQ[#2] && PrimeQ[#3] && #2 <= #3 &); Table[{n, t[n, c]}, {n, 1, 30}]
Join[{0}, Table[Length[t[n, c]], {n, 1, 130}]]
(* Peter J. C. Moses, Mar 29 2026 *)

Prog

(PARI) apply( {A394741(n, c=0)=n>6&&forprime(p=2, sqrtint(n-4), bigomega(n-p^2)==2 && c++); c}, [0..99])

Crossrefs

Cf. A393710, A394740, A394742, A395255.

Sequence in context: A178670 A026921 A198568 * A327310 A232539 A157044

Adjacent sequences: A394738 A394739 A394740 * A394742 A394743 A394744

Keyword

nonn

Author

Clark Kimberling, Apr 09 2026

Status

approved