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A046897 Sum of divisors of n that are not divisible by 4.

%I A046897 #69 Aug 03 2025 13:04:11
%S A046897 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,
%T A046897 42,40,24,30,72,32,3,48,54,48,39,38,60,56,18,42,96,44,36,78,72,48,12,
%U A046897 57,93,72,42,54,120,72,24,80,90,60,72,62,96,104,3,84,144,68,54,96,144,72
%N A046897 Sum of divisors of n that are not divisible by 4.
%C A046897 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A046897 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
%D A046897 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.
%D A046897 G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island 2002, p. 133.
%D A046897 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Oxford, Fifth edition, 1979, p. 314.
%D A046897 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.
%D A046897 C. J. Moreno and S. S. Wagstaff, Jr., Sums of Squares of Integers, Chapman & Hall, 2006.
%H A046897 Reinhard Zumkeller, <a href="/A046897/b046897.txt">Table of n, a(n) for n = 1..10000</a>
%H A046897 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>.
%H A046897 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>.
%F A046897 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]
%F A046897 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
%F A046897 Expansion of (1 - phi(q)^4) / 8 in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Jan 25 2008
%F A046897 Equals inverse Mobius transform of A190621. - _Gary W. Adamson_, Jul 03 2008
%F A046897 A000118(n) = 8*a(n) for all n>0.
%F A046897 Dirichlet g.f.: (1 - 4^(1-s)) * zeta(s) * zeta(s-1). - _Michael Somos_, Oct 21 2015
%F A046897 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
%F A046897 From _Peter Bala_, Dec 19 2021: (Start)
%F A046897 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)) )
%F A046897 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)
%F A046897 Sum_{k=1..n} a(k) ~ (Pi^2/16) * n^2. - _Amiram Eldar_, Oct 04 2022
%e A046897 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 + ...
%p A046897 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
%t A046897 a[n_] := Sum[ Boole[ !Divisible[d, 4]]*d, {d, Divisors[n]}]; Table[ a[n], {n, 1, 71}] (* _Jean-François Alcover_, Dec 12 2011 *)
%t A046897 DivisorSum[#1, # &, Mod[#, 4] != 0 &] & /@ Range[71] (* _Jayanta Basu_, Jun 30 2013 *)
%t A046897 a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 - 1) / 8, {q, 0, n}]; (* _Michael Somos_, Dec 30 2014 *)
%t A046897 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 *)
%o A046897 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, if(d%4, d)))};
%o A046897 (Magma) A := Basis( ModularForms( Gamma0(4), 2), 72); B<q> := (A[1] - 1)/8 + A[2]; B; /* _Michael Somos_, Dec 30 2014 */
%o A046897 (Haskell)
%o A046897 a046897 1 = 1
%o A046897 a046897 n = product $ zipWith
%o A046897             (\p e -> if p == 2 then 3 else div (p ^ (e + 1) - 1) (p - 1))
%o A046897             (a027748_row n) (a124010_row n)
%o A046897 -- _Reinhard Zumkeller_, Aug 12 2015
%Y A046897 Cf. A000203, A000118, A051731, A069733, A027748, A124010, A190621, A000593 (not divis. by 2), A046913 (not divis. by 3), A116073 (not divis. by 5).
%K A046897 nonn,mult
%O A046897 1,2
%A A046897 _N. J. A. Sloane_