OFFSET
1,3
COMMENTS
First occurrence of k: 2, 1, 3, 9, 15, 64, 45, 256, 96, 144, 192, 4096, 240, ????, 768, 576, 480, ????, 720, ..., . See A246063. - Robert G. Wilson v, Oct 31 2013
a(n) is the number of pairs (u, v) in NxZ satisfying u^2-v^2=n. See Kühleitner. - Michel Marcus, Jul 30 2017
REFERENCES
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000
M. Kühleitner, An Omega Theorem on Differences of Two Squares, Acta Mathematica Universitatis Comenianae, Vol. 61, 1 (1992) pp. 117-123. See Lemma 1 p. 2.
John Shareshian and Sheila Sundaram, Ramanujan sums and rectangular power sums, arXiv:2305.12007 [math.CO], 2023. Mentions this sequence.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
FORMULA
Multiplicative with a(2^e) = e-1 if e>0, a(p^e) = 1+e if p>2.
G.f.: Sum_{k>0} x^k / (1 - (-x)^k) = Sum_{k>0} -(-x)^k / (1 + (-x)^k).
Möbius transform is period 4 sequence [ 1, -1, 1, 1, ...].
G.f.: Sum_{k>=1} x^(k^2) * (1+x^(2*k))/(1-x^(2*k)). - Joerg Arndt, Nov 08 2010
a(4*n + 2) = 0. a(n) = -(-1)^n * A048272(n). a(2*n - 1) = A099774(n). a(4*n) = A000005(n). a(4*n + 1) = A000005(4*n + 1). a(4*n - 1) = 2 * A078703(n).
a(n) = A094572(n) / 2. - Ray Chandler, Aug 23 2014
Bisection: a(2*k-1) = A000005(2*k-1), a(2*k) = A183063(2*k) - A001227(2*k), k >= 1. See the Hardy reference, p. 142 where a(n) = sigma^*_0(n). - Wolfdieter Lang, Jan 07 2017
a(n) = d(n) - 2*d(n/2) + 2*d(n/4) where d(n) = 0 if n is not an integer. See Kühleitner.
a(n) = Sum_{d|n} [(d mod 2) = (n/d mod 2)], where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Mar 21 2022
From Amiram Eldar, Nov 29 2022: (Start)
Dirichlet g.f.: zeta(s)^2*(1 + 2^(1-2*s) - 2^(1-s)).
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (2*gamma-1)*n/2, where gamma is Euler's constant (A001620). (End)
a(n) = (-1)^n * Sum_{d|2*n} cos(d*Pi/2). - Ridouane Oudra, Sep 27 2024
EXAMPLE
x + 2*x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^8 + 3*x^9 + 2*x^11 + 2*x^12 + ...
MAPLE
f:= proc(n) if n::odd then numtheory:-tau(n) elif n mod 4 = 0 then numtheory:-tau(n/4) else 0 fi end proc;
seq(f(i), i=1..100); # Robert Israel, Aug 24 2014
MATHEMATICA
Rest[ CoefficientList[ Series[ Sum[x^k/(1 - (-x)^k), {k, 111}], {x, 0, 110}], x]] (* Robert G. Wilson v, Sep 20 2005 *)
Table[If[OddQ[n], DivisorSigma[0, n], If[OddQ[n/2], 0, DivisorSigma[0, n/4]]], {n, 100} ] (* Ray Chandler, Aug 23 2014 *)
PROG
(PARI) {a(n) = if( n<1, 0, (-1)^n * sumdiv( n, d, (-1)^d))}
(PARI) {a(n) = if( n<1, 0, if( n%2, numdiv(n), if( n%4, 0, numdiv(n/4))))} /* Michael Somos, Sep 02 2006 */
(PARI) d(n) = if (denominator(n)==1, numdiv(n), 0);
a(n) = numdiv(n) - 2*d(n/2) + 2*d(n/4); \\ Michel Marcus, Jul 30 2017
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Michael Somos, Sep 04 2005
STATUS
approved