A326401 - OEIS

%I #23 Dec 24 2022 19:03:44

%S 1,1,3,3,4,3,8,5,9,4,10,9,14,8,12,11,16,9,20,12,24,10,22,15,21,14,27,

%T 24,28,12,32,21,30,16,32,27,38,20,42,20,40,24,44,30,36,22,46,33,57,21,

%U 48,42,52,27,40,40,60,28,58,36,62,32,72,43,56,30,68,48,66,32

%N Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

%H Amiram Eldar, <a href="/A326401/b326401.txt">Table of n, a(n) for n = 1..10000</a>

%H Claudia Rella, <a href="https://arxiv.org/abs/2212.10606">Resurgence, Stokes constants, and arithmetic functions in topological string theory</a>, arXiv:2212.10606 [hep-th], 2022. See pages 21 - 23.

%F a(n) = Sum_{d|n, n/d==1 (mod 3)} d - Sum_{d|n, n/d==2 (mod 3)} d.

%F a(n) = A326399(n) - A326400(n).

%F Multiplicative with a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1)/(p - 1) if p == 1 (mod 3), and (p^(e+1) + (-1)^e)/(p + 1) if p == 2 (mod 3). - _Amiram Eldar_, Oct 25 2020

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2) * Product_{primes p == 2 (mod 3)} 1/(1 + 1/p^2) = (1/2) * A175646 * (2*Pi^2/27)/A340577 = 0.3906512064... . - _Amiram Eldar_, Nov 06 2022

%t nmax = 70; CoefficientList[Series[Sum[k x^k/(1 + x^k + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

%t Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 3]] &] - DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 70}]

%t f[p_, e_] := Which[p == 3, p^e, Mod[p, 3] == 1, (p^(e + 1) - 1)/(p - 1), Mod[p, 3] == 2, (p^(e + 1) + (-1)^e)/(p + 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 25 2020 *)

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 3, 3^f[i,2], if(f[i,1]%3 == 1, (f[i,1]^(f[i,2]+1) - 1)/(f[i,1] - 1), (f[i,1]^(f[i,2]+1) + (-1)^f[i,2])/(f[i,1] + 1))));} \\ _Amiram Eldar_, Nov 06 2022

%Y Cf. A000593, A002324, A050469, A078708, A326399, A326400.

%Y Cf. A175646, A340577.

%K nonn,mult,easy

%O 1,3

%A _Ilya Gutkovskiy_, Sep 11 2019