A394740 a(n) = number of triples (x, y, z) such that x^2 + y*z = n, where x, y, z are distinct primes and y < z.

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

Offset

0,20

Comments

Also, number of primes p not dividing n such that n - p^2 is a semiprime but not a square. - M. F. Hasler, Apr 11 2026

Links

Table of n, a(n) for n=0..85.

Example

a(19) = 2 counts these triples: (2, 3, 5), (3, 2, 5).

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 &&
     DuplicateFreeQ[{#1, #2, #3}] &);
Join[{0}, Table[Length[t[n, c]], {n, 1, 130}]]
(* Peter J. C. Moses, Mar 29 2026 *)

Prog

(PARI) apply( {A394740(n, c=0)=n>6&&forprime(p=2, sqrtint(n-6), n%p&& bigomega(n-p^2)==2 &&!issquare(n-p^2)&&c++); c}, [0..99])

Crossrefs

Cf. A393710, A394741, A394742.

Sequence in context: A325562 A176917 A302110 * A085983 A285701 A088183

Adjacent sequences: A394737 A394738 A394739 * A394741 A394742 A394743

Keyword

nonn

Author

Clark Kimberling, Apr 09 2026

Status

approved