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A058344 Difference between the sum of the odd aliquot divisors of n and the sum of the even aliquot divisors of n.

%I A058344 #18 Feb 25 2018 23:00:40
%S A058344 0,1,1,-1,1,2,1,-5,4,4,1,-8,1,6,9,-13,1,5,1,-10,11,10,1,-28,6,12,13,
%T A058344 -12,1,6,1,-29,15,16,13,-29,1,18,17,-38,1,10,1,-16,33,22,1,-68,8,19,
%U A058344 21,-18,1,14,17,-48,23,28,1,-60,1,30,41,-61,19,18,1,-22,27,22,1,-97,1,36,49,-24,19,22,1,-94,40
%N A058344 Difference between the sum of the odd aliquot divisors of n and the sum of the even aliquot divisors of n.
%C A058344 The number of terms where the sum of the odd parts is greater than the sum of the even parts up to 10^n: 6, 57, 521, 5070, 50223, 500707, 5002236, ...
%H A058344 Antti Karttunen, <a href="/A058344/b058344.txt">Table of n, a(n) for n = 1..65537</a>
%F A058344 G.f.: Sum_{k>0} -(-1)^k * k * x^(2*k) / (1 - x^k). - _Michael Somos_, Aug 21 2005
%e A058344 a(28) = -12 because the sum of the even divisors of 28 (2, 4 and 14) = 20 and the sum of the odd divisors of 28 (1 and 7) = 8.
%e A058344 G.f. = x^2 + x^3 - x^4 + x^5 + 2*x6 + x^7 - 5*x^8 + 4*x^9 + 4*x^10 + x^11 + ...
%t A058344 f[n_Integer] := Block[{d = Most[Divisors[n]]}, Plus @@ (-d*(-1)^d)]; Table[ f[n], {n, 81}] (* or *)
%t A058344 Rest[ CoefficientList[ Series[ Sum[ -(-1)^k*k*x^(2k)/(1 - x^k), {k, 100}], {x, 0, 81}], x]] (* _Robert G. Wilson v_, Aug 26 2005 *)
%t A058344 Table[With[{d=Most[Divisors[n]]},Total[Select[d,OddQ]]-Total[Select[d,EvenQ]]],{n,90}] (* _Harvey P. Dale_, Feb 16 2013 *)
%o A058344 (PARI) {a(n) = if(n<1, 0, sumdiv(n, d, (d<n)* d * -(-1)^d))}; /* _Michael Somos_, Aug 21 2005 */
%o A058344 (PARI) {a(n) = if( n<1, 0, polcoeff( sum(k=1, n\2, -(-1)^k * k * x^(2*k) / (1 - x^k), x * O(x^n)), n))}; /* _Michael Somos_, Aug 21 2005 */
%Y A058344 Cf. A002129.
%K A058344 sign,look
%O A058344 1,6
%A A058344 _Robert G. Wilson v_, Dec 14 2000
%E A058344 Signs added by _Michael Somos_, Aug 21 2005