OFFSET
1,2
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The o.g.f. is (theta_3(0,x)^4 - 1)/8, see the Hardy reference, eqs. 9.2.1, 9.2.3 and 9.2.4 on p. 133 for Sum' m*u_m. Also Hardy-Wright, p. 314. See also the Somos, Jan 25 2008 formula below. - Wolfdieter Lang, Dec 11 2016
REFERENCES
J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 194.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island 2002, p. 133.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth edition, 1979, p. 314.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 31, Article 273.
C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall, 2006.
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
FORMULA
a(n) = (-1)^(n+1)*Sum_{d divides n} (-1)^(n/d+d)*d. Multiplicative with a(2^e) = 3, a(p^e) = (p^(e+1)-1)/(p-1) for an odd prime p. - Vladeta Jovovic, Sep 10 2002 [For a proof of the multiplicative property, see for example Moreno and Wagstaff, p. 33. - N. J. A. Sloane, Nov 09 2016]
G.f.: Sum_{k>0} x^k/(1+(-x)^k)^2, or Sum_{k>0} k*x^k/(1+(-x)^k). - Vladeta Jovovic, Dec 16 2002
Expansion of (1 - phi(q)^4) / 8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Jan 25 2008
Equals inverse Mobius transform of A190621. - Gary W. Adamson, Jul 03 2008
A000118(n) = 8*a(n) for all n>0.
Dirichlet g.f.: (1 - 4^(1-s)) * zeta(s) * zeta(s-1). - Michael Somos, Oct 21 2015
L.g.f.: log(Product_{k>=1} (1 - x^(4*k))/(1 - x^k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 14 2018
From Peter Bala, Dec 19 2021: (Start)
Logarithmic g.f.: Sum_{n >= 1} a(n)*x^n/n = Sum_{n >= 1} x^n*(1 + x^n + x^(2*n))/( n*(1 - x^(4*n)) )
G.f.: Sum_{n >= 1} x^n*(x^(6*n) + 2*x^(5*n) + 3*x^(4*n) + 3*x^(2*n) + 2*x^n + 1)/(1 - x^(4*n))^2. (End)
Sum_{k=1..n} a(k) ~ (Pi^2/16) * n^2. - Amiram Eldar, Oct 04 2022
EXAMPLE
G.f. = q + 3*q^2 + 4*q^3 + 3*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 3*q^8 + 13*q^9 + ...
MAPLE
A046897 := proc(n) if n mod 4 = 0 then numtheory[sigma](n)-4*numtheory[sigma](n/4) ; else numtheory[sigma](n) ; end if; end proc: # R. J. Mathar, Mar 23 2011
MATHEMATICA
a[n_] := Sum[ Boole[ !Divisible[d, 4]]*d, {d, Divisors[n]}]; Table[ a[n], {n, 1, 71}] (* Jean-François Alcover, Dec 12 2011 *)
DivisorSum[#1, # &, Mod[#, 4] != 0 &] & /@ Range[71] (* Jayanta Basu, Jun 30 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 - 1) / 8, {q, 0, n}]; (* Michael Somos, Dec 30 2014 *)
f[2, e_] := 3; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)
PROG
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, if(d%4, d)))};
(Magma) A := Basis( ModularForms( Gamma0(4), 2), 72); B<q> := (A[1] - 1)/8 + A[2]; B; /* Michael Somos, Dec 30 2014 */
(Haskell)
a046897 1 = 1
a046897 n = product $ zipWith
(\p e -> if p == 2 then 3 else div (p ^ (e + 1) - 1) (p - 1))
(a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Aug 12 2015
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
STATUS
approved