A094572 Number of pairs of integers x, y (of either sign) with x^2 - y^2 = n.

2, 0, 4, 2, 4, 0, 4, 4, 6, 0, 4, 4, 4, 0, 8, 6, 4, 0, 4, 4, 8, 0, 4, 8, 6, 0, 8, 4, 4, 0, 4, 8, 8, 0, 8, 6, 4, 0, 8, 8, 4, 0, 4, 4, 12, 0, 4, 12, 6, 0, 8, 4, 4, 0, 8, 8, 8, 0, 4, 8, 4, 0, 12, 10, 8, 0, 4, 4, 8, 0, 4, 12, 4, 0, 12, 4, 8, 0, 4, 12, 10, 0, 4, 8, 8, 0, 8, 8, 4, 0, 8, 4, 8, 0, 8, 16, 4, 0, 12, 6

Offset: 1

Author

N. J. A. Sloane, Nov 02 2008

Keywords

• nonn
• easy

Comments

The old entry with this sequence number was a duplicate of A058071.

a(n) == 2 (mod 4) if n is a square otherwise a(n) is divisible by 4. Cf. A112329. - Peter Bala, Jan 08 2025

References

• M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).

Crossrefs

Cf. A000005, A001620, A093061, A112329.

Programs

Maple

with(numtheory); f:=proc(n) if n mod 2 = 1 then RETURN(2*tau(n)); fi; if n mod 4 = 0 then RETURN(2*tau(n/4)); fi; 0; end;

Mathematica

Table[If[OddQ[n],2DivisorSigma[0,n],If[OddQ[n/2],0,2DivisorSigma[0,n/4]]],{n,100}] (* Ray Chandler, Aug 23 2014 *)

PARI

a(n) = if(n%2, 2 * numdiv(n), if(n % 4 == 0, 2 * numdiv(n/4), 0)); \\ Amiram Eldar, Apr 13 2024

Formula

a(n) = 2*d(n) if n is odd, = 2*d(n/4) if n == 0 mod 4, otherwise 0, where d() = A000005().

a(n) = 2 * A112329(n). - Ray Chandler, Aug 23 2014

From Amiram Eldar, Apr 13 2024: (Start)

Dirichlet g.f.: 2*zeta(s)^2*(1 + 2^(1-2*s) - 2^(1-s)).

Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma-1)*n, where gamma is Euler's constant (A001620). (End)