cp's OEIS Frontend

A215947 Difference between the sum of the even divisors and the sum of the odd divisors of 2n.

%I A215947 #59 Sep 26 2023 13:25:00
%S A215947 1,5,4,13,6,20,8,29,13,30,12,52,14,40,24,61,18,65,20,78,32,60,24,116,
%T A215947 31,70,40,104,30,120,32,125,48,90,48,169,38,100,56,174,42,160,44,156,
%U A215947 78,120,48,244,57,155,72,182,54,200,72,232,80,150,60,312,62,160
%N A215947 Difference between the sum of the even divisors and the sum of the odd divisors of 2n.
%C A215947 Multiplicative because a(n) = -A002129(2*n), A002129 is multiplicative and a(1) = -A002129(2) = 1. - _Andrew Howroyd_, Jul 31 2018
%H A215947 Michel Lagneau, <a href="/A215947/b215947.txt">Table of n, a(n) for n = 1..10000</a>
%H A215947 Hartosh Singh Bal and Gaurav Bhatnagar, <a href="https://arxiv.org/abs/2102.10804">Two curious congruences for the sum of divisors function</a>, arXiv:2102.10804 [math.NT], 2021. Mentions this sequence.
%H A215947 H. Movasati and Y. Nikdelan, <a href="http://arxiv.org/abs/1603.09411">Gauss-Manin Connection in Disguise: Dwork Family</a>, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2017.
%F A215947 From _Andrew Howroyd_, Jul 28 2018: (Start)
%F A215947 a(n) = 4*sigma(n) - sigma(2*n).
%F A215947 a(n) = -A002129(2*n). (End)
%F A215947 G.f.: Sum_{k>=1} x^k*(1 + 4*x^k + x^(2*k))/(1 - x^(2*k))^2. - _Ilya Gutkovskiy_, Sep 14 2019
%F A215947 a(p) = p + 1 for p prime >= 3. - _Bernard Schott_, Sep 14 2019
%F A215947 a(n) = A239050(n) - A062731(n) - _Omar E. Pol_, Mar 06 2021 (after _Andrew Howroyd_)
%F A215947 From _Amiram Eldar_, Nov 18 2022: (Start)
%F A215947 Multiplicative with a(2^e) = 2^(e+2) - 3, and a(p^e) = sigma(p^e) = (p^(e+1) - 1)/(p-1) for p > 2.
%F A215947 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). (End)
%F A215947 Dirichlet g.f.: zeta(s)*zeta(s-1)*(1+2^(1-s)). - _Amiram Eldar_, Jan 05 2023
%F A215947 From _Peter Bala_, Sep 25 2023: (Start)
%F A215947 a(2*n) = sigma(2*n) + 2*sigma(n); a(2*n+1) = sigma(2*n+1) = A008438(n)
%F A215947 G.f.: A(q) = Sum_{n >= 1} n*q^n*(1 + 3*q^n)/(1 - q^(2*n)).
%F A215947 Logarithmic g.f.: Sum_{n >= 1} a(n)*q^n/n = Sum_{n >= 1} log(1/(1 - q^n)) + Sum_{n >= 1} log(1/(1 - q^(2*n))) = log (G(q)), where G(q) is the g.f. of A002513. (End)
%e A215947 a(6) = 20 because the divisors of 2*6 = 12 are {1, 2, 3, 4, 6, 12} and (12 + 6 + 4 +2) - (3 + 1) = 20.
%p A215947 with(numtheory):for n from 1 to 100 do:x:=divisors(2*n):n1:=nops(x):s0:=0:s1:=0:for m from 1 to n1 do: if irem(x[m],2)=0 then s0:=s0+x[m]:else s1:=s1+x[m]:fi:od:if s0>s1  then printf(`%d, `,s0-s1):else fi:od:
%t A215947 a[n_] := DivisorSum[2n, (1 - 2 Mod[#, 2]) #&];
%t A215947 Array[a, 62] (* _Jean-François Alcover_, Sep 13 2018 *)
%t A215947 edod[n_]:=Module[{d=Divisors[2n]},Total[Select[d,EvenQ]]-Total[ Select[ d,OddQ]]]; Array[edod,70] (* _Harvey P. Dale_, Jul 30 2021 *)
%o A215947 (PARI) a(n) = 4*sigma(n) - sigma(2*n); \\ _Andrew Howroyd_, Jul 28 2018
%Y A215947 Cf. A000593, A002129, A022998 (Moebius transform), A074400, A195382, A195690.
%Y A215947 Cf. A002513, A062731, A111003, A239050.
%K A215947 nonn,look,mult
%O A215947 1,2
%A A215947 _Michel Lagneau_, Aug 28 2012