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 A046897 Sum of divisors of n that are not divisible by 4. 31
 1, 3, 4, 3, 6, 12, 8, 3, 13, 18, 12, 12, 14, 24, 24, 3, 18, 39, 20, 18, 32, 36, 24, 12, 31, 42, 40, 24, 30, 72, 32, 3, 48, 54, 48, 39, 38, 60, 56, 18, 42, 96, 44, 36, 78, 72, 48, 12, 57, 93, 72, 42, 54, 120, 72, 24, 80, 90, 60, 72, 62, 96, 104, 3, 84, 144, 68, 54, 96, 144, 72 (list; graph; refs; listen; history; text; internal format)
 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 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 (A051731) of "count, 4*n = 0": (1, 2, 3, 0, 5, 6, 7, 0,...). - 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 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 *) PROG (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, if(d%4, d)))}; (MAGMA) A := Basis( ModularForms( Gamma0(4), 2), 72); B := (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 Cf. A000118, A051731, A069733. Cf. A027748, A124010. Sequence in context: A073181 A183100 * A109506 A000113 A069915 A033634 Adjacent sequences:  A046894 A046895 A046896 * A046898 A046899 A046900 KEYWORD nonn,mult AUTHOR STATUS approved

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Last modified December 17 06:45 EST 2018. Contains 318192 sequences. (Running on oeis4.)