A321141 - OEIS

%I #26 Jul 29 2023 06:43:06

%S 1,6,29,291,3127,48246,823545,16909060,387459858,10019533302,

%T 285311670613,8920489178073,302875106592255,11113363271736486,

%U 437893951444713443,18447307036548136965,827240261886336764179,39346708467688595378892,1978419655660313589123981

%N a(n) = Sum_{d|n} sigma_n(d).

%H Seiichi Manyama, <a href="/A321141/b321141.txt">Table of n, a(n) for n = 1..386</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F a(n) = [x^n] Sum_{k>=1} sigma_n(k)*x^k/(1 - x^k).

%F a(n) = Sum_{d|n} d^n*tau(n/d).

%F a(n) ~ n^n. - _Vaclav Kotesovec_, Feb 16 2020

%p with(numtheory): seq(coeff(series(add(sigma[n](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 20); # _Muniru A Asiru_, Oct 28 2018

%t Table[Sum[DivisorSigma[n, d], {d, Divisors[n]}] , {n, 19}]

%t Table[SeriesCoefficient[Sum[DivisorSigma[n, k] x^k/(1 - x^k), {k, 1, n}], {x, 0, n}], {n, 19}]

%o (PARI) a(n) = sumdiv(n, d, sigma(d, n)); \\ _Michel Marcus_, Oct 28 2018

%o (Python)

%o from sympy import divisor_sigma, divisors

%o def A321141(n):

%o     return sum(divisor_sigma(d,0)*(n//d)**n for d in divisors(n,generator=True)) # _Chai Wah Wu_, Feb 15 2020

%o (Magma) [&+[DivisorSigma(n, d):d in Divisors(n)]:n in [1..20]]; // _Vincenzo Librandi_, Feb 16 2020

%Y Cf. A000005, A007429, A007433, A023887, A319194, A320940, A321140.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Oct 28 2018