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

1, 5, 4, 13, 6, 20, 8, 29, 13, 30, 12, 52, 14, 40, 24, 61, 18, 65, 20, 78, 32, 60, 24, 116, 31, 70, 40, 104, 30, 120, 32, 125, 48, 90, 48, 169, 38, 100, 56, 174, 42, 160, 44, 156, 78, 120, 48, 244, 57, 155, 72, 182, 54, 200, 72, 232, 80, 150, 60, 312, 62, 160

Offset

1,2

Comments

Multiplicative because a(n) = -A002129(2*n), A002129 is multiplicative and a(1) = -A002129(2) = 1. - Andrew Howroyd, Jul 31 2018

Links

Michel Lagneau, Table of n, a(n) for n = 1..10000

Hartosh Singh Bal and Gaurav Bhatnagar, Two curious congruences for the sum of divisors function, arXiv:2102.10804 [math.NT], 2021. Mentions this sequence.

H. Movasati and Y. Nikdelan, Gauss-Manin Connection in Disguise: Dwork Family, arXiv preprint arXiv:1603.09411 [math.AG], 2016-2017.

Formula

From Andrew Howroyd, Jul 28 2018: (Start)

a(n) = 4*sigma(n) - sigma(2*n).

a(n) = -A002129(2*n). (End)

G.f.: Sum_{k>=1} x^k*(1 + 4*x^k + x^(2*k))/(1 - x^(2*k))^2. - Ilya Gutkovskiy, Sep 14 2019

a(p) = p + 1 for p prime >= 3. - Bernard Schott, Sep 14 2019

a(n) = A239050(n) - A062731(n) - Omar E. Pol, Mar 06 2021 (after Andrew Howroyd)

From Amiram Eldar, Nov 18 2022: (Start)

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.

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/8 = 1.2337005... (A111003). (End)

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1+2^(1-s)). - Amiram Eldar, Jan 05 2023

From Peter Bala, Sep 25 2023: (Start)

a(2*n) = sigma(2*n) + 2*sigma(n); a(2*n+1) = sigma(2*n+1) = A008438(n)

G.f.: A(q) = Sum_{n >= 1} n*q^n*(1 + 3*q^n)/(1 - q^(2*n)).

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)

Example

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.

Maple

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:

Mathematica

a[n_] := DivisorSum[2n, (1 - 2 Mod[#, 2]) #&];
Array[a, 62] (* Jean-François Alcover, Sep 13 2018 *)
edod[n_]:=Module[{d=Divisors[2n]}, Total[Select[d, EvenQ]]-Total[ Select[ d, OddQ]]]; Array[edod, 70] (* Harvey P. Dale, Jul 30 2021 *)

Prog

(PARI) a(n) = 4*sigma(n) - sigma(2*n); \\ Andrew Howroyd, Jul 28 2018

Crossrefs

Cf. A000593, A002129, A022998 (Moebius transform), A074400, A195382, A195690.

Cf. A002513, A062731, A111003, A239050.

Sequence in context: A338157 A189235 A019068 * A226555 A147685 A353151

Adjacent sequences: A215944 A215945 A215946 * A215948 A215949 A215950

Keyword

nonn,look,mult

Author

Michel Lagneau, Aug 28 2012

Status

approved