A316961 -id:A316961 - cp's OEIS frontend

A316962 Expansion of Product_{k>=1} (1 + sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

1, 1, 3, 7, 11, 25, 51, 87, 129, 286, 462, 760, 1312, 2102, 3470, 5988, 8840, 13884, 22577, 33545, 55961, 85341, 126705, 194317, 293621, 435040, 641472, 971503, 1462483, 2108161, 3124489, 4474579, 6545809, 9561923, 13518678, 19809034, 28387625, 40286631, 57039233

Offset: 0

Ilya Gutkovskiy, Jul 17 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000203, A180305, A192065, A279786, A316961.

with(numtheory): a:=series(mul(1+sigma(k)*x^k,k=1..100),x=0,39): seq(coeff(a,x,n),n=0..38); # Paolo P. Lava, Apr 02 2019

nmax = 38; CoefficientList[Series[Product[(1 + DivisorSigma[1, k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 38; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) DivisorSigma[1, j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d DivisorSigma[1, d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 38}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*sigma(j)^k*x^(j*k)/k).

A319111 Expansion of Product_{k>=1} 1/(1 - phi(k)*x^k), where phi = Euler totient function (A000010).

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 22, 38, 63, 105, 174, 278, 447, 707, 1122, 1766, 2729, 4213, 6482, 9880, 15069, 22799, 34290, 51378, 76777, 114365, 169324, 250162, 368505, 540575, 792042, 1154798, 1680385, 2439101, 3530308, 5103380, 7349875, 10564955, 15155752, 21696072, 31007949, 44199845

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..8200

Cf. A000010, A061255, A159929, A318917.

Cf. A279784, A306327, A316961.

with(numtheory): a:=series(mul(1/(1-phi(k)*x^k),k=1..50),x=0,42): seq(coeff(a,x,n),n=0..41); # Paolo P. Lava, Apr 02 2019

nmax = 41; CoefficientList[Series[Product[1/(1 - EulerPhi[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[Exp[Sum[Sum[EulerPhi[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d EulerPhi[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 41}]

G.f.: exp(Sum_{k>=1} Sum_{j>=1} phi(j)^k*x^(j*k)/k).
From Vaclav Kotesovec, Feb 08 2019: (Start)
a(n) ~ c * 2^(2*n/5), where
c = 18827.6460615531202942792897255332975807324818737172163... if mod(n,5) = 0
c = 18827.5079339024144115146595255453426552477117955925738... if mod(n,5) = 1
c = 18827.4967567108036710998657106724179082561779712900405... if mod(n,5) = 2
c = 18827.4818413568083742650057347700058389606441225811016... if mod(n,5) = 3
c = 18827.4547665561882994953942505862213438332903500716893... if mod(n,5) = 4
(End)

cp's OEIS Frontend

A316962 Expansion of Product_{k>=1} (1 + sigma(k)*x^k), where sigma(k) is the sum of the divisors of k (A000203).

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