A217872 - OEIS

%I #31 Nov 16 2020 03:07:27

%S 1,9,64,2401,7776,2985984,2097152,2562890625,10604499373,

%T 3570467226624,743008370688,232218265089212416,793714773254144,

%U 21035720123168587776,504857282956046106624,727423121747185263828481,2185911559738696531968,43567528752021332753202420081

%N a(n) = sigma(n)^n.

%C Here sigma(n) = A000203(n) is the sum of the divisors of n.

%C Compare to A023887(n) = sigma(n,n).

%H Paul D. Hanna, <a href="/A217872/b217872.txt">Table of n, a(n) for n = 1..200</a>

%F Logarithmic derivative of A156217.

%F From _Amiram Eldar_, Nov 16 2020: (Start)

%F Sum_{n>=1} 1/a(n) = A215140.

%F Sum_{n>=1} (-1)^(n+1)/a(n) = A215141. (End)

%e L.g.f.: L(x) = x + 3^2*x^2/2 + 4^3*x^3/3 + 7^4*x^4/4 + 6^5*x^5/5 + 12^6*x^6/6 +...

%e where exponentiation yields the g.f. of A156217:

%e exp(L(x)) = 1 + x + 5*x^2 + 26*x^3 + 634*x^4 + 2273*x^5 + 502568*x^6 +...

%t Table[DivisorSigma[1, n]^n, {n, 1, 20}] (* _Amiram Eldar_, Nov 16 2020 *)

%o (PARI) {a(n)=sigma(n)^n}

%o for(n=1,20,print1(a(n),", "))

%Y Cf. A000203, A023887, A156217, A215140, A215141.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 01 2012