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%I #55 Jun 12 2022 02:57:54
%S 1,1,4,5,6,4,8,13,13,6,12,20,14,8,24,29,18,13,20,30,32,12,24,52,31,14,
%T 40,40,30,24,32,61,48,18,48,65,38,20,56,78,42,32,44,60,78,24,48,116,
%U 57,31,72,70,54,40,72,104,80,30,60,120,62,32,104,125,84,48,68,90,96,48,72
%N Absolute difference between sum of odd divisors of n and sum of even divisors of n.
%C The generating function equals 1/8 at q = Lambda = 0.1076539192... (A072558) the "One-Ninth" constant. - _Michael Somos_, Jul 21 2006
%C Absolute value of A002129. - _John W. Layman_, Sep 27 2012
%C The Möbius transform is 1, 0, 3, 4, 5, 0, 7, 8, 9, 0, 11, 12, 13, 0, 15, 16, 17, 0, 19, 20, 21, 0, 23, 24, 25, 0, 27, ... - _R. J. Mathar_, Jan 08 2013
%D G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.
%H G. C. Greubel, <a href="/A113184/b113184.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/One-NinthConstant.html">One-Ninth Constant</a>.
%F Multiplicative with a(2^e) = 2^(e+1)-3 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p>2.
%F G.f.: Sum_{k>0} -(-x)^k/(1+(-x)^k)^2 = Sum_{k>0} k*x^k/(1-(-x)^k).
%F Expansion of (1-(2/Pi)^2(2E(k)-K(k))K(k))/8 in powers of nome q where E(k) and K(k) are complete elliptic integrals and q=exp(-Pi*K(k')/K(k)). - _Michael Somos_, Jul 21 2006
%F Bisection: a(2*k-1) = A000203(2*k-1), a(2*k) = A146076(2*k) - A000593(2*k), k >= 1. See the Hardy reference where a(n) = sigma^*_1(n). - _Wolfdieter Lang_, Jan 07 2017
%F From _Peter Bala_, Dec 11 2020: (Start)
%F a(n) = Sum_{d | n, d != 2 (mod 4)} d.
%F O.g.f.: Sum_{k >= 1, k != 2 (mod 4)} k*x^k/(1 - x^k). Cf. A284362.
%F Define a(n) = 0 for n < 1. Then a(n) = e(n) + a(n-1) + a(n-3) - a(n-6) - a(n-10) + + - -, where [1, 3, 6, 10, ...] is the sequence of triangular numbers A000217, and e(n) = (-1)^(n+1)*n if n is a triangular number; otherwise e(n) = 0. Examples of this recurrence are given below. (End)
%F Dirichlet g.f.: Sum_{n>0} a(n)/n^s = zeta(s) * zeta(s-1) * (1+2^(3-3*s)) / (1+2^(1-s)). - _Werner Schulte_, Jan 23 2021
%F Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 16. - _Vaclav Kotesovec_, Aug 20 2021
%e From _Peter Bala_, Dec 11 2020: (Start)
%e n = 15: n is a triangular number, so e(n) = (-1)^(n+1)*n = 15 and a(15) = 15 + a(14) + a(12) - a(9) - a(5) = 15 + 8 + 20 - 13 - 6 = 24;
%e n = 16: n is a not triangular number, so e(n) = 0 and a(16) = a(15) + a(13) - a(10) - a(6) + a(1) = 24 + 14 - 6 - 4 + 1 = 29. (End)
%t f[n_]:=Module[{dn=Divisors[n],odn,edn},odn=Select[dn,OddQ];edn=Select[dn,EvenQ];Abs[Total[odn]-Total[edn]]]
%t f/@Range[80] (* _Harvey P. Dale_, Feb 25 2011 *)
%t max = 80; s = (1/x)*Sum[k*x^k/(1 - (-x)^k), {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* _Jean-François Alcover_, Dec 04 2015 *)
%t f[p_, e_] := If[p == 2, 2^(e + 1) - 3, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jun 12 2022 *)
%o (PARI) a(n)=if(n<1, 0, (-1)^n*sumdiv(n,d,(-1)^d*d))
%o (PARI) {a(n)=local(A,p,e); if(n<1, 0, A=factor(n); prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, 2^(e+1)-3, (p^(e+1)-1)/(p-1)))))}
%Y Cf. A002129(n) = -(-1)^n a(n).
%Y Cf. A072558, A284362.
%K nonn,mult
%O 1,3
%A _Michael Somos_, Oct 17 2005
%E Name corrected by _Wolfdieter Lang_, Jan 07 2017
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