A340577 -id:A340577 - cp's OEIS frontend

A326401 Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

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, 24, 28, 12, 32, 21, 30, 16, 32, 27, 38, 20, 42, 20, 40, 24, 44, 30, 36, 22, 46, 33, 57, 21, 48, 42, 52, 27, 40, 40, 60, 28, 58, 36, 62, 32, 72, 43, 56, 30, 68, 48, 66, 32

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000
Claudia Rella, Resurgence, Stokes constants, and arithmetic functions in topological string theory, arXiv:2212.10606 [hep-th], 2022. See pages 21 - 23.

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

Cf. A175646, A340577.

nmax = 70; CoefficientList[Series[Sum[k x^k/(1 + x^k + x^(2 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 3]] &] - DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 70}]
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 *)

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

a(n) = Sum_{d|n, n/d==1 (mod 3)} d - Sum_{d|n, n/d==2 (mod 3)} d.
a(n) = A326399(n) - A326400(n).
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
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

A340857 Decimal expansion of constant K5 = 29log(2+sqrt(5))(Product_{primes p == 1 (mod 5)} (1-4(2p-1)/(p(p+1)^2)))/(15Pi^2).

Original entry on oeis.org

2, 6, 2, 6, 5, 2, 1, 8, 8, 7, 2, 0, 5, 3, 6, 7, 6, 6, 6, 7, 5, 9, 6, 2, 0, 1, 1, 4, 7, 2, 0, 8, 8, 3, 4, 6, 5, 3, 0, 2, 0, 4, 3, 9, 3, 0, 6, 4, 7, 4, 4, 7, 3, 9, 1, 0, 6, 8, 2, 5, 5, 1, 0, 5, 8, 7, 0, 9, 2, 6, 6, 8, 3, 8, 6, 9, 0, 2, 2, 7, 4, 1, 7, 9, 4, 1, 9, 3, 8, 3, 6, 5, 5, 2, 3, 5, 0, 0, 2, 0, 1, 0, 0, 8, 9, 1

Offset: 0

Artur Jasinski, Jan 24 2021

Finch and Sebah, 2009, p. 7 (see link) call this constant K_5. K_5 is related to the Mertens constant C(5,1) (see A340839). For more references see the links in A340711. Finch and Sebah give the following definition:

Consider the asymptotic enumeration of m-th order primitive Dirichlet characters mod n. Let b_m(n) denote the count of such characters. There exists a constant 0 < K_m < oo such that Sum_{n <= N} b_m(n) ∼ K_m*N*log(N)^(d(m) - 2) as N -> oo, where d(m) is the number of divisors of m.

			0.262652188720536766675962011472088346530204393064744739106825510587...

Steven Finch and Pascal Sebah, Residue of a Mod 5 Euler Product, arXiv:0912.3677 [math.NT], 2009 p. 10.

Cf. A340878 (K3), A340879 (K4).
Cf. A340004, A340794, A340665, A340127.
Cf. A340629, A340710, A340711, A340628.
Cf. A175646, A301429, A333240.
Cf. A175647, A248930, A248938, A335963.
Cf. A340576, A340577, A340578, A334826.

$MaxExtraPrecision = 1000; digits = 121; f[p_] := (1 - 4*(2*p-1)/(p*(p+1)^2));
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]]*S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*P[5, 1, m]; sump = sump + difp; PrintTemporary[m]; m++];
RealDigits[Chop[N[29*Log[2+Sqrt[5]]/(15*Pi^2) * Exp[sump], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 25 2021, took over 50 minutes *)

Equals (29/25)*(Product_{primes p} (1-1/p)^2*(1+gcd(p-1,5)/(p-1))) [Finch and Sebah, 2009, p. 10].

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A326401 Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

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A340857 Decimal expansion of constant K5 = 29log(2+sqrt(5))(Product_{primes p == 1 (mod 5)} (1-4(2p-1)/(p(p+1)^2)))/(15Pi^2).

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cp's OEIS Frontend

A326401 Expansion of Sum_{k>=1} k * x^k / (1 + x^k + x^(2*k)).

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Author

Keywords

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Crossrefs

Programs

Mathematica

PARI

Formula

A340857 Decimal expansion of constant K5 = 29*log(2+sqrt(5))*(Product_{primes p == 1 (mod 5)} (1-4*(2*p-1)/(p*(p+1)^2)))/(15*Pi^2).

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A340857 Decimal expansion of constant K5 = 29log(2+sqrt(5))(Product_{primes p == 1 (mod 5)} (1-4(2p-1)/(p(p+1)^2)))/(15Pi^2).