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A094572
Number of pairs of integers x, y (of either sign) with x^2 - y^2 = n.
5
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
1,1
COMMENTS
The old entry with this sequence number was a duplicate of A058071.
REFERENCES
M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 236.
LINKS
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
FORMULA
a(n) = 2d(n) if n is odd, = 2d(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)
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 *)
PROG
(PARI) a(n) = if(n%2, 2 * numdiv(n), if(n % 4 == 0, 2 * numdiv(n/4), 0)); \\ Amiram Eldar, Apr 13 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 02 2008
STATUS
approved