A288571 -id:A288571 - cp's OEIS frontend

A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij))^(1/(ij)).

1, 1, 2, 10, 34, 218, 1708, 12556, 97340, 1139932, 12602584, 142757624, 1983086488, 26745019000, 402951386576, 7181178238672, 115410887636752, 2039658743085584, 42354537803172640, 815690033731561888, 17593347085888752416, 416765224159172991136, 9379433694333768563392

Offset: 0

Ilya Gutkovskiy, Aug 31 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..446
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000005, A107742, A168243, A280541, A288571, A318695, A318977.

seq(n!*coeff(series(mul((1+x^k)^(tau(k)/k),k=1..100),x=0,23),x,n),n=0..22); # Paolo P. Lava, Jan 09 2019

nmax = 22; CoefficientList[Series[Product[Product[(1 + x^(i j))^(1/(i j)), {i, 1, nmax}], {j, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Product[(1 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 22; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) DivisorSigma[0, d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 22}]
nmax = 22; s = 1 + x; Do[s *= Sum[Binomial[DivisorSigma[0, k]/k, j]*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; Take[CoefficientList[s, x], nmax + 1] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 01 2018 *)

E.g.f.: Product_{k>=1} (1 + x^k)^(tau(k)/k), where tau = number of divisors (A000005).

E.g.f.: exp(Sum_{k>=1} ( Sum_{d|k} (-1)^(k/d+1)*tau(d) ) * x^k/k).

A318768 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} tau(j), where tau = number of divisors (A000005).

Original entry on oeis.org

1, 2, 4, 2, 4, 8, 4, 0, 10, 8, 4, 8, 4, 8, 16, -5, 4, 20, 4, 8, 16, 8, 4, 0, 10, 8, 20, 8, 4, 32, 4, -14, 16, 8, 16, 20, 4, 8, 16, 0, 4, 32, 4, 8, 40, 8, 4, -20, 10, 20, 16, 8, 4, 40, 16, 0, 16, 8, 4, 32, 4, 8, 40, -28, 16, 32, 4, 8, 16, 32, 4, 0, 4, 8, 40, 8, 16, 32, 4

Offset: 1

Ilya Gutkovskiy, Sep 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000005, A007425, A007426, A051062 (positions of 0's), A288571.

Cf. A001620, A082633.

Table[Sum[(-1)^(n/d + 1) Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[n]}], {n, 79}]
nmax = 79; Rest[CoefficientList[Series[Sum[DivisorSum[k, DivisorSigma[0, #] &] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 79; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSum[k, DivisorSigma[0, #] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := If[p == 2, 1 + (7 - e^2)*e/6, Binomial[e + 3, 3]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)

a(n) = sumdiv(n, d, (-1)^(n/d+1) * sumdiv(d, j, numdiv(j))); \\ Michel Marcus, Sep 04 2018

G.f.: Sum_{k>=1} tau_3(k)*x^k/(1 + x^k), where tau_3() = A007425.
L.g.f.: log(Product_{k>=1} (1 + x^k)^(tau_3(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
Multiplicative with a(2^e) = 1 + (7-e^2)*e/6, and a(p^e) = binomial(e+3,3) for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Dec 18 2023: (Start)
Dirichlet g.f.: zeta(s)^4 * (1 - 1/2^(s-1)).
Sum_{k=1..n} a(k) ~ (log(2)/2) * n * (log(n)^2 + (8 * gamma - log(2) - 2) * log(n) + 12 * gamma^2 - 8 * gamma + log(2) + 2 - 4 * gamma * log(2) + log(2)^2/3 - 8 * gamma_1), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). (End)

A317531 Expansion of Sum_{p prime, k>=1} x^(p^k)/(1 + x^(p^k)).

Original entry on oeis.org

0, 1, 1, 0, 1, 0, 1, -1, 2, 0, 1, -1, 1, 0, 2, -2, 1, -1, 1, -1, 2, 0, 1, -2, 2, 0, 3, -1, 1, -1, 1, -3, 2, 0, 2, -2, 1, 0, 2, -2, 1, -1, 1, -1, 3, 0, 1, -3, 2, -1, 2, -1, 1, -2, 2, -2, 2, 0, 1, -2, 1, 0, 3, -4, 2, -1, 1, -1, 2, -1, 1, -3, 1, 0, 3, -1, 2, -1, 1, -3, 4, 0, 1, -2, 2, 0, 2, -2, 1, -2, 2, -1, 2, 0, 2

Offset: 1

Ilya Gutkovskiy, Jul 30 2018

sign

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A001222, A048272, A069513, A246655, A288571, A305614, A317528.

nmax = 95; Rest[CoefficientList[Series[Sum[Boole[PrimePowerQ[k]] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 95; Rest[CoefficientList[Series[Log[Product[(1 + Boole[PrimePowerQ[k]] x^k)^(1/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
Table[DivisorSum[n, (-1)^(n/# + 1) &, PrimePowerQ[#] &], {n, 95}]

A317531(n) = sumdiv(n,d,((-1)^(n/d+1))*(1==omega(d))); \\ Antti Karttunen, Sep 30 2018

G.f.: Sum_{k>=1} x^A246655(k)/(1 + x^A246655(k)).
L.g.f.: log(Product_{p prime, k>=1} (1 + x^(p^k))^(1/p^k)) = Sum_{n>=1} a(n)*x^n/n.
a(n) = Sum_{d|n} (-1)^(n/d+1)*A069513(d).
If n is odd, a(n) = A001222(n).

A141844 Expansion of (1+x)(1+x^2)/((1-x)^2(1+x+x^2)(1-4x)).

Original entry on oeis.org

1, 6, 27, 113, 458, 1839, 7365, 29470, 117891, 471577, 1886322, 7545303, 30181229, 120724934, 482899755, 1931599041, 7726396186, 30905584767, 123622339093, 494489356398, 1977957425619, 7911829702505, 31647318810050, 126589275240231, 506357100960957

Offset: 0

Gary W. Adamson, Jul 11 2008

nonn

Old name was: a(n) = 4*a(n-1) + A042968(n), with a(0) = 1, where A042968 = "not divisible by 4": (1, 2, 3, 5, 6, 7, 9, 10, 11, ...). After the correction of a(13) the definition could be simplified. - N. J. A. Sloane, Aug 23 2018

			a(3) = 4*a(2) + A042968(3) = 4*27 + 5 = 113.
a(13) = 4*a(12) + A042968(13) = 4*30181229 + 18 = 120724934.

Michael De Vlieger, Table of n, a(n) for n = 0..1660
Index entries for linear recurrences with constant coefficients, signature (5, -4, 1, -5, 4).

Cf. A042968.

I:=[1,6,27,113,458]; [n le 5 select I[n] else 5*Self(n-1)-4*Self(n-2)+Self(n-3)-5*Self(n-4)+4*Self(n-5): n in [1..30]]; // Vincenzo Librandi, Jun 28 2017, Jul 28 2018

CoefficientList[Series[(1 + x) (1 + x^2) / ((1 - x)^2 (1 + x + x^2) (1 - 4 x)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 28 2017 *)
Nest[Append[#, Block[{k = #[[-1, -1]] + 1}, While[Mod[k, 4] == 0, k++]; {4 #[[-1, 1]] + k, k}]] &, {{1, 1}}, 24][[All, 1]] (* Michael De Vlieger, Jun 30 2018 *)
LinearRecurrence[{5, -4, 1, -5, 4}, {1, 6, 27, 113, 458}, 25] (* Robert G. Wilson v, Jul 28 2018 *)

Vec((1 + x)*(1 + x^2) / ((1 - x)^2*(1 - 4*x)*(1 + x + x^2)) + O(x^30)) \\ Colin Barker, Jun 26 2017

a(n) = 5*a(n-1) - 4*a(n-2) + a(n-3) - 5*a(n-4) + 4*a(n-5) for n>4. - Colin Barker, Jun 26 2017

Corrected by Charlie Neder, Jun 22 2018

Edited by N. J. A. Sloane, Aug 23 2018, merging old entry A288571 with this one.

A327242 Expansion of Sum_{k>=1} tau(k) * x^k / (1 + x^k)^2, where tau = A000005.

Original entry on oeis.org

1, 0, 5, -5, 7, 0, 9, -18, 18, 0, 13, -25, 15, 0, 35, -47, 19, 0, 21, -35, 45, 0, 25, -90, 38, 0, 58, -45, 31, 0, 33, -108, 65, 0, 63, -90, 39, 0, 75, -126, 43, 0, 45, -65, 126, 0, 49, -235, 66, 0, 95, -75, 55, 0, 91, -162, 105, 0, 61, -175, 63, 0, 162, -233, 105

Offset: 1

Ilya Gutkovskiy, Sep 14 2019

Inverse Moebius transform of A002129.

Dirichlet convolution of A000005 with A181983.

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

Cf. A000005, A002129, A007429, A008586 (positions of negative terms), A016825 (positions of 0's), A181983, A288417, A288571.

[&+[(-1)^(d+1)*d*#Divisors(n div d):d in Divisors(n)]:n in [1..65]]; // Marius A. Burtea, Sep 14 2019

nmax = 65; CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/(1 + x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
a[n_] := Sum[(-1)^(d + 1) d DivisorSigma[0, n/d], {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]
f[p_, e_] := (p^(e + 2) - (e + 2)*p + e + 1)/(p-1)^2; f[2, e_] := 3*e + 5 - 2^(e+2); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* Amiram Eldar, May 25 2025 *)

a(n) = {sumdiv(n, d, (-1)^(d + 1) * d * numdiv(n/d))} \\ Andrew Howroyd, Sep 14 2019

a(n) = Sum_{d|n} A002129(d).
a(n) = Sum_{d|n} (-1)^(d + 1) * d * tau(n/d).
Multiplicative with a(2^e) = 3*e + 5 - 2^(e+2), and a(p^e) = (p^(e+2) - (e+2)*p +e + 1)/(p-1)^2 for an odd prime p. - Amiram Eldar, May 25 2025

A369100 Dirichlet g.f.: zeta(s)^3 * (1 - 2^(1-s))^2.

Original entry on oeis.org

1, -1, 3, -2, 3, -3, 3, -2, 6, -3, 3, -6, 3, -3, 9, -1, 3, -6, 3, -6, 9, -3, 3, -6, 6, -3, 10, -6, 3, -9, 3, 1, 9, -3, 9, -12, 3, -3, 9, -6, 3, -9, 3, -6, 18, -3, 3, -3, 6, -6, 9, -6, 3, -10, 9, -6, 9, -3, 3, -18, 3, -3, 18, 4, 9, -9, 3, -6, 9, -9, 3, -12, 3, -3, 18

Offset: 1

Vaclav Kotesovec, Jan 13 2024

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A007425, A048272, A288571.

Table[Sum[Sum[-(-1)^d, {d, Divisors[k]}]*(-1)^(n/k+1), {k, Divisors[n]}], {n, 1, 100}]
f[p_, e_] := (e + 1)*(e + 2)/2; f[2, e_] := (e^2 - 5*e + 2)/2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 13 2024 *)

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e=f[i,2]; if(p == 2, (e^2-5*e+2)/2, (e+1)*(e+2)/2));} \\ Amiram Eldar, Jan 13 2024

Sum_{k=1..n} a(k) ~ n * log(2)^2.

Multiplicative with a(2^e) = (e^2-5*e+2)/2, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Jan 13 2024

cp's OEIS Frontend

A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(i*j))^(1/(i*j)).

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A318768 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} tau(j), where tau = number of divisors (A000005).

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A317531 Expansion of Sum_{p prime, k>=1} x^(p^k)/(1 + x^(p^k)).

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A141844 Expansion of (1+x)*(1+x^2)/((1-x)^2*(1+x+x^2)*(1-4*x)).

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A327242 Expansion of Sum_{k>=1} tau(k) * x^k / (1 + x^k)^2, where tau = A000005.

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A369100 Dirichlet g.f.: zeta(s)^3 * (1 - 2^(1-s))^2.

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A318696 Expansion of e.g.f. Product_{i>=1, j>=1} (1 + x^(ij))^(1/(ij)).

A141844 Expansion of (1+x)(1+x^2)/((1-x)^2(1+x+x^2)(1-4x)).