A326401 -id:A326401 - cp's OEIS frontend

A326399 Expansion of Sum_{k>=1} k * x^k / (1 - x^(3*k)).

1, 2, 3, 5, 5, 6, 8, 10, 9, 11, 11, 15, 14, 16, 15, 21, 17, 18, 20, 27, 24, 23, 23, 30, 26, 28, 27, 40, 29, 33, 32, 42, 33, 35, 40, 45, 38, 40, 42, 55, 41, 48, 44, 57, 45, 47, 47, 63, 57, 57, 51, 70, 53, 54, 56, 80, 60, 59, 59, 81, 62, 64, 72, 85, 70, 69, 68, 87, 69, 88

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001817, A002131, A016777, A050460, A078181, A078708, A326400, A326401.

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

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

G.f.: Sum_{k>0} x^(3*k-2) / (1 - x^(3*k-2))^2. - Seiichi Manyama, Jun 29 2023

A326400 Expansion of Sum_{k>=1} k * x^(2k) / (1 - x^(3k)).

Original entry on oeis.org

0, 1, 0, 2, 1, 3, 0, 5, 0, 7, 1, 6, 0, 8, 3, 10, 1, 9, 0, 15, 0, 13, 1, 15, 5, 14, 0, 16, 1, 21, 0, 21, 3, 19, 8, 18, 0, 20, 0, 35, 1, 24, 0, 27, 9, 25, 1, 30, 0, 36, 3, 28, 1, 27, 16, 40, 0, 31, 1, 45, 0, 32, 0, 42, 14, 39, 0, 39, 3, 56, 1, 45, 0, 38, 15, 40, 8, 42, 0, 71

Offset: 1

Ilya Gutkovskiy, Sep 11 2019

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A001822, A002131, A016789, A078182, A078708, A326399, A326401.

Cf. A050464, A363900.

nmax = 80; CoefficientList[Series[Sum[k x^(2 k)/(1 - x^(3 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{2}, Mod[n/#, 3]] &], {n, 1, 80}]

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

G.f.: Sum_{k>0} x^(3*k-1) / (1 - x^(3*k-1))^2. - Seiichi Manyama, Jun 29 2023

A326503 Expansion of Sum_{k>=1} x^k * (1 - x^(2k)) / (1 + x^k + x^(2k))^2.

Original entry on oeis.org

1, -1, 1, 3, -4, -1, 8, -5, 1, 4, -10, 3, 14, -8, -4, 11, -16, -1, 20, -12, 8, 10, -22, -5, 21, -14, 1, 24, -28, 4, 32, -21, -10, 16, -32, 3, 38, -20, 14, 20, -40, -8, 44, -30, -4, 22, -46, 11, 57, -21, -16, 42, -52, -1, 40, -40, 20, 28, -58, -12, 62, -32, 8, 43, -56, 10

Offset: 1

Ilya Gutkovskiy, Sep 12 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 33 - 35.

Cf. A002129, A050457, A078181, A078182, A078708, A162397 (Moebius transform), A326401.

nmax = 66; CoefficientList[Series[Sum[x^k (1 - x^(2 k))/(1 + x^k + x^(2 k))^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[#, 3]] &] - DivisorSum[n, # &, MemberQ[{2}, Mod[#, 3]] &], {n, 1, 66}]
f[p_, e_] := If[Mod[p, 3] == 1, (p^(e + 1) - 1)/(p - 1), ((-p)^(e + 1) - 1)/(-p - 1)]; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 28 2023 *)

a(n)={sumdiv(n, d, d*((d+1)%3-1))} \\ Andrew Howroyd, Sep 12 2019

a(n) = Sum_{d|n, d==1 (mod 3)} d - Sum_{d|n, d==2 (mod 3)} d.
a(n) = A078181(n) - A078182(n).
Multiplicative with a(3^e) = 1, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1 (mod 3) and a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 2 (mod 3). - Amiram Eldar, Nov 28 2023

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2k)) / (1 + x^(2k) + x^(4*k)).

Original entry on oeis.org

1, 2, 3, 4, 4, 6, 8, 8, 9, 8, 10, 12, 14, 16, 12, 16, 16, 18, 20, 16, 24, 20, 22, 24, 21, 28, 27, 32, 28, 24, 32, 32, 30, 32, 32, 36, 38, 40, 42, 32, 40, 48, 44, 40, 36, 44, 46, 48, 57, 42, 48, 56, 52, 54, 40, 64, 60, 56, 58, 48, 62, 64, 72, 64, 56, 60

Offset: 1

Ilya Gutkovskiy, Sep 12 2019

			G.f. = x + 2*x^2 + 3*x^3 + 4*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 8*x^8 + ... - _Michael Somos_, Oct 23 2019

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003586 (fixed points), A035178, A050469, A122373, A326401.

Cf. A175646, A340578.

nmax = 66; CoefficientList[Series[Sum[k x^k (1 + x^(2 k))/(1 + x^(2 k) + x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, # &, MemberQ[{1}, Mod[n/#, 6]] &] - DivisorSum[n, # &, MemberQ[{5}, Mod[n/#, 6]] &], {n, 1, 66}]
f[p_, e_] := Which[p < 5, p^e, Mod[p, 6] == 5, (p^(e + 1) - (-1)^(e + 1))/(p + 1), Mod[p, 6] == 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Dec 02 2020 *)

a(n) = { sumdiv(n, d, d*((n/d%6==1)-(n/d%6==5))) } \\ Andrew Howroyd, Sep 12 2019

{a(n) = if( n<0, 0, sumdiv( n, d, n/d * kronecker( -12, d)))}; /* Michael Somos, Oct 23 2019 */

a(n) = Sum_{d|n, n/d==1 (mod 6)} d - Sum_{d|n, n/d==5 (mod 6)} d.
G.f.: Sum_{k>=0}  x^(6*k+1) / (1 - x^(6*k+1))^2 - x^(6*k+5) / (1 - x^(6*k+5))^2. - Michael Somos, Oct 23 2019
Multiplicative with a(p^e) = p^e if p < 5, (p^(e+1)-(-1)^(e+1))/(p+1) if p == 5 (mod 6), and (p^(e+1)-1)/(p-1) if p == 1 (mod 6). - Amiram Eldar, Dec 02 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{primes p == 5 (mod 6)} 1/(1+1/p^2) * Product_{primes p == 1 (mod 3)} 1/(1 - 1/p^2) = A340578 * A175646 / 2 = 0.48831400806... . - Amiram Eldar, Nov 06 2022

cp's OEIS Frontend

A326399 Expansion of Sum_{k>=1} k * x^k / (1 - x^(3*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A326400 Expansion of Sum_{k>=1} k * x^(2*k) / (1 - x^(3*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A326503 Expansion of Sum_{k>=1} x^k * (1 - x^(2*k)) / (1 + x^k + x^(2*k))^2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2*k)) / (1 + x^(2*k) + x^(4*k)).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A326400 Expansion of Sum_{k>=1} k * x^(2k) / (1 - x^(3k)).

A326503 Expansion of Sum_{k>=1} x^k * (1 - x^(2k)) / (1 + x^k + x^(2k))^2.

A326575 Expansion of Sum_{k>=1} k * x^k * (1 + x^(2k)) / (1 + x^(2k) + x^(4*k)).