A023881 -id:A023881 - cp's OEIS frontend

A158952 Inverse Euler transform of the number of partitions in expanding space (A023881).

1, 2, 9, 67, 625, 7903, 117649, 2105342, 43048905, 1000976352, 25937424601, 743191207969, 23298085122481, 793763217701693, 29192928060852217, 1152939097060278256, 48661191875666868481, 2185919903971766191000

Offset: 1

Paul D. Hanna, Mar 31 2009

nonn

			Let G(x) = Sum_{n>=0} A023881(n)*x^n then
G(x) = 1 + x + 3*x^2 + 12*x^3 + 82*x^4 + 725*x^5 + 8811*x^6 +...
G(x) = 1/[(1-x)*(1-x^2)^2*(1-x^3)^9*(1-x^4)^67*(1-x^5)^625*...].

Cf. A023881, A158947.

Table[Sum[DivisorSigma[d, d]*MoebiusMu[n/d], {d, Divisors[n]}]/n, {n, 1, 20}] (* Vaclav Kotesovec, Oct 09 2019 *)

{a(n)=(1/n)*sumdiv(n,d,sigma(d,d)*moebius(n/d))}

a(n) = (1/n)*Sum_{d|n} sigma(d,d)*moebius(n/d).

a(n) ~ n^(n-1). - Vaclav Kotesovec, Oct 09 2019

A023887 a(n) = sigma_n(n): sum of n-th powers of divisors of n.

Original entry on oeis.org

1, 5, 28, 273, 3126, 47450, 823544, 16843009, 387440173, 10009766650, 285311670612, 8918294543346, 302875106592254, 11112685048647250, 437893920912786408, 18447025552981295105, 827240261886336764178, 39346558271492178925595, 1978419655660313589123980

Offset: 1

Olivier Gérard

Logarithmic derivative of A023881.

Compare to A217872(n) = sigma(n)^n.

			The divisors of 6 are 1, 2, 3 and 6, so a(6) = 1^6 + 2^6 + 3^6 + 6^6 = 47450.

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 38.

Nick Hobson, Table of n, a(n) for n = 1..100

Cf. A000203, A001157-A001160, A013954-A013972, A023881, A199858, subdiagonal in A109974.

A023887 := proc(n)
    numtheory[sigma][n](n) ;
end proc:
seq(A023887(n),n=1..10) ; # R. J. Mathar, Apr 06 2022

Table[DivisorSigma[n,n],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Feb 26 2009 *)

makelist(divsum(n,n),n,1,20); /* Emanuele Munarini, Mar 26 2011 */

a(n) = sigma(n,n); \\ Nick Hobson, Nov 25 2006

from sympy import divisor_sigma
def A023887(n): return divisor_sigma(n,n) # Chai Wah Wu, Jun 19 2022

G.f.: Sum_{n>0} (n*x)^n/(1-(n*x)^n). - Vladeta Jovovic, Oct 27 2002
From Nick Hobson, Nov 25 2006: (Start)
If the canonical prime factorization of n > 1 is the product of p^e(p) then sigma_n(n) = Product_p ((p^(n*(e(p)+1)))-1)/(p^n-1).
sigma_n(n) is odd if and only if n is a square or twice a square. (End)
Conjecture: sigma_m(n) = sigma(n^m * rad(n)^(m-1))/sigma(rad(n)^(m-1)) for n > 0 and m > 0, where sigma = A000203 and rad = A007947. - Velin Yanev, Aug 24 2017
a(n) ~ n^n. - Vaclav Kotesovec, Nov 02 2018
Sum_{n>=1} 1/a(n) = A199858. - Amiram Eldar, Nov 19 2020

Edited by N. J. A. Sloane, Nov 25 2006

A205814 G.f.: Product_{n>=1} [ (1 - 2^nx^n) / (1 - (n+2)^nx^n) ]^(1/n).

Original entry on oeis.org

1, 1, 9, 54, 482, 4239, 55561, 785554, 14133055, 285547760, 6666380256, 172748192767, 4974178683908, 156462697434990, 5354832107694444, 197710292330150160, 7839473395324929677, 332071887435037103895, 14968498613432649146050, 715294449027151380463781

Offset: 0

Paul D. Hanna, Feb 01 2012

nonn

Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.

			G.f.: A(x) = 1 + x + 9*x^2 + 54*x^3 + 482*x^4 + 4239*x^5 + 55561*x^6 +...
where the g.f. equals the product:
A(x) = (1-2*x)/(1-3*x) * ((1-2^2*x^2)/(1-4^2*x^2))^(1/2) * ((1-2^3*x^3)/(1-5^3*x^3))^(1/3) * ((1-2^4*x^4)/(1-6^4*x^4))^(1/4) * ((1-2^5*x^5)/(1-7^5*x^5))^(1/5) *...
The logarithm equals the l.g.f. of A205815:
log(A(x)) = x + 17*x^2/2 + 136*x^3/3 + 1585*x^4/4 + 16986*x^5/5 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A205815 (log), A205811, A023881.

{a(n)=polcoeff(exp(sum(m=1, n+1, x^m/m*sum(k=1, m, binomial(m, k)*sigma(m, k)*2^(m-k))+x*O(x^n))), n)}

{a(n)=polcoeff(prod(k=1, n, ((1-2^k*x^k)/(1-(k+2)^k*x^k +x*O(x^n)))^(1/k)), n)}

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k) * sigma(n,k) * 2^(n-k) ).

a(n) ~ exp(2) * n^(n-1). - Vaclav Kotesovec, Oct 08 2016

A186633 G.f.: Product_{n>=1} (1 + n^n*x^n)^(1/n).

Original entry on oeis.org

1, 1, 2, 11, 71, 705, 8470, 127284, 2231342, 45505041, 1048341387, 27059612615, 771505041184, 24109351520196, 818862556900962, 30044548635160841, 1184027932446520895, 49883835880381353808, 2237276988838875420087

Offset: 0

Paul D. Hanna, Mar 19 2011

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 71*x^4 + 705*x^5 +...
A(x) = (1 + x) *(1 + 4*x^2)^(1/2) *(1 + 27*x^3)^(1/3) *(1 + 256*x^4)^(1/4) *...

Vaclav Kotesovec, Table of n, a(n) for n = 0..370

Cf. variant: A023881.

{a(n)=if(n<0, 0, polcoeff(prod(k=1, n, (1+k^k*x^k+x*O(x^n))^(1/k)), n))}

a(n) ~ n^(n-1). - Vaclav Kotesovec, Nov 06 2014

A205811 G.f.: Product_{n>=1} [ (1 - x^n) / (1 - (n+1)^n*x^n) ]^(1/n).

Original entry on oeis.org

1, 1, 6, 29, 221, 1897, 23502, 335334, 5923570, 119354491, 2758647259, 71079498533, 2031108928680, 63520842121792, 2161164726505952, 79394066773371245, 3133259427956392983, 132166451829847198316, 5934636812034634649249, 282609413111134846839482

Offset: 0

Paul D. Hanna, Feb 01 2012

nonn

Here sigma(n,k) equals the sum of the k-th powers of the divisors of n.

			G.f.: A(x) = 1 + x + 6*x^2 + 29*x^3 + 221*x^4 + 1897*x^5 + 23502*x^6 +...
where the g.f. equals the product:
A(x) = (1-x)/(1-2*x) * ((1-x^2)/(1-3^2*x^2))^(1/2) * ((1-x^3)/(1-4^3*x^3))^(1/3) * ((1-x^4)/(1-5^4*x^4))^(1/4) * ((1-x^5)/(1-6^5*x^5))^(1/5) *...
The logarithm equals the l.g.f. of A205812:
log(A(x)) = x + 11*x^2/2 + 70*x^3/3 + 719*x^4/4 + 7806*x^5/5 + 122534*x^6/6 +...

Cf. A205812 (log), A205814, A023881.

{a(n)=polcoeff(exp(sum(m=1,n+1,x^m/m*sum(k=1,m,binomial(m,k)*sigma(m,k))+x*O(x^n))),n)}

{a(n)=polcoeff(prod(k=1,n,((1-x^k)/(1-(k+1)^k*x^k +x*O(x^n)))^(1/k)),n)}

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{k=1..n} binomial(n,k) * sigma(n,k) ).

A158095 G.f.: A(x) = exp( Sum_{n>=1} 2sigma(n,n-1)x^n/n ).

Original entry on oeis.org

1, 2, 5, 14, 61, 370, 3454, 40880, 614346, 10848514, 222870183, 5175100204, 134514302384, 3859406052466, 121274242936381, 4139268759072626, 152532132931199062, 6034430112251517608, 255114747410233804986

Offset: 0

Paul D. Hanna, Mar 20 2009

nonn

			G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 61*x^4 + 370*x^5 +...

Cf. A023881, A294645.

a(n)=polcoeff(exp(sum(m=1,n,2*sigma(m,m-1)*x^m/m)+x*O(x^n)),n)

A163203 G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)d^n] x^n/n ).

Original entry on oeis.org

1, 1, 2, 11, 79, 713, 8486, 127372, 2248390, 45527161, 1048442107, 27060812167, 771886991408, 24110090108332, 818871809076474, 30044771201925569, 1184069354974499199, 49884064948928968400, 2237283630465903060711

Offset: 0

Paul D. Hanna, Jul 22 2009

nonn

A variant of A023881, which is defined by g.f.:
exp( Sum_{n>=1} [Sum_{d|n} d^n] * x^n/n )
where A023881 is the number of partitions in expanding space.
Compare also to the g.f. of A006950 given by:
exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d] * x^n/n ),
where A006950(n) is the number of partitions of n in which each even part occurs with even multiplicity.

			G.f.: 1 + x + 2*x^2 + 11*x^3 + 79*x^4 + 713*x^5 + 8486*x^6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..380

Cf. A023881, A006950, A002129.

{a(n)=polcoeff(exp(sum(m=1, n+1, sumdiv(m, d, (-1)^(m-d)*d^m)*x^m/m)+x*O(x^n)), n)}

a(n) ~ n^(n-1) * (1 + exp(-1)/n + (3*exp(-1)/2 + 2*exp(-2))/n^2). - Vaclav Kotesovec, Aug 17 2015

A294948 Expansion of Product_{n>=1} (1 - n^n*x^n)^(1/n).

Original entry on oeis.org

1, -1, -2, -7, -57, -541, -7126, -108072, -1966034, -40620681, -952305757, -24824933859, -714742428220, -22491627743504, -768696164146118, -28344822040761041, -1121925480573229737, -47442205907345238412, -2134679753840086267669

Offset: 0

Seiichi Manyama, Nov 11 2017

sign

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1/n, g(n) = n^n.

Seiichi Manyama, Table of n, a(n) for n = 0..387

Column k=0 of A294947.

Cf. A023881, A023887.

N=66; x='x+O('x^N); Vec(prod(k=1, N, (1-k^k*x^k)^(1/k)))

G.f.: exp(-Sum_{k>0} A023887(k)*x^k/k).

a(0) = 1 and a(n) = -(1/n) * Sum_{k=1..n} A023887(k)*a(n-k) for n > 0.

A156360 G.f.: A(x) = exp( Sum_{n>=1} sigma_n(2n)x^n/n ), where sigma_n(2n) is the sum of the n-th powers of the divisors of 2n.

Original entry on oeis.org

1, 3, 15, 120, 1450, 25383, 591563, 17156364, 595635903, 24023004840, 1102221504614, 56652798990909, 3222918574782830, 200989079661549750, 13632214370613131094, 998992560620311541814, 78653794343072884416393

Offset: 0

Paul D. Hanna, Feb 08 2009

nonn

			G.f.: A(x) = 1 + 3*x + 15*x^2 + 120*x^3 + 1450*x^4 + 25383*x^5 +...
log(A(x)) = 3*x + 21*x^2/2 + 252*x^3/3 + 4369*x^4/4 + 103158*x^5/5 +...
sigma(2n,n) = [3,21,252,4369,103158,3037530,106237176,4311810305,...].

Cf. variant: A023881 (number of partitions in expanding space).

Cf. A179504.

{a(n)=polcoeff(exp(sum(k=1,n,sigma(2*k,k)*x^k/k,x*O(x^n))),n)}

{a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(2*k,k)*a(n-k)))}

a(n) = (1/n)*Sum_{k=1..n} sigma(2*k,k)*a(n-k) for n>0, with a(0) = 1.

a(n) ~ 2^n * n^(n-1). - Vaclav Kotesovec, Oct 31 2024

A294946 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of exp(Sum_{j>0} sigma_k(j)*x^j/j) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 5, 12, 1, 1, 9, 32, 82, 1, 1, 17, 90, 304, 725, 1, 1, 33, 260, 1162, 3537, 8811, 1, 1, 65, 762, 4516, 17435, 52010, 128340, 1, 1, 129, 2252, 17722, 86529, 310193, 895397, 2257687, 1, 1, 257, 6690, 69964, 431675, 1865766, 6286826, 18016416, 45658174

Offset: 0

Seiichi Manyama, Nov 11 2017

			Square array begins:
     1,    1,     1,     1,      1, ...
     1,    1,     1,     1,      1, ...
     3,    5,     9,    17,     33, ...
    12,   32,    90,   260,    762, ...
    82,  304,  1162,  4516,  17722, ...
   725, 3537, 17435, 86529, 431675, ...

Columns k=0..2 give A023881, A023882, A294813.
Rows n=0+1, 2 give A000012, A000051(n+1).
Cf. A294296, A294947.

G.f. of column k: Product_{j>0} 1/(1 - j^j*x^j)^(j^(k-1)).

cp's OEIS Frontend

A158952 Inverse Euler transform of the number of partitions in expanding space (A023881).

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A023887 a(n) = sigma_n(n): sum of n-th powers of divisors of n.

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A205814 G.f.: Product_{n>=1} [ (1 - 2^n*x^n) / (1 - (n+2)^n*x^n) ]^(1/n).

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A186633 G.f.: Product_{n>=1} (1 + n^n*x^n)^(1/n).

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A205811 G.f.: Product_{n>=1} [ (1 - x^n) / (1 - (n+1)^n*x^n) ]^(1/n).

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A158095 G.f.: A(x) = exp( Sum_{n>=1} 2*sigma(n,n-1)*x^n/n ).

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A163203 G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)*d^n] * x^n/n ).

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A205814 G.f.: Product_{n>=1} [ (1 - 2^nx^n) / (1 - (n+2)^nx^n) ]^(1/n).

A158095 G.f.: A(x) = exp( Sum_{n>=1} 2sigma(n,n-1)x^n/n ).

A163203 G.f.: exp( Sum_{n>=1} [Sum_{d|n} (-1)^(n-d)d^n] x^n/n ).

A156360 G.f.: A(x) = exp( Sum_{n>=1} sigma_n(2n)x^n/n ), where sigma_n(2n) is the sum of the n-th powers of the divisors of 2n.