A158095 -id:A158095 - cp's OEIS frontend

A294645 a(n) = Sum_{d|n} d^(n+1).

1, 9, 82, 1057, 15626, 282252, 5764802, 134480385, 3486843451, 100048830174, 3138428376722, 107006334784468, 3937376385699290, 155572843119354936, 6568408508343827972, 295150156996346511361, 14063084452067724991010, 708236696816416252145973

Offset: 1

Seiichi Manyama, Nov 05 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Divisor Function

Column k=1 of A308504.

Cf. A023882, A023887, A082245, A158095, A292312.

Table[DivisorSigma[n + 1, n], {n, 1, 20}] (* Vaclav Kotesovec, Oct 07 2020 *)

PARI
```
{a(n) = sigma(n, n+1)}
```

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

N=20; x='x+O('x^N); Vec(x*deriv(-log(prod(k=1, N, 1-(k*x)^k)))) \\ Seiichi Manyama, Jun 02 2019

G.f.: Sum_{k>0} k^(k+1)*x^k/(1-(k*x)^k).
L.g.f.: -log(Product_{k>=1} (1 - (k*x)^k)) = Sum_{k>=1} a(k)*x^k/k. - Seiichi Manyama, Jun 02 2019
a(n) ~ n^(n+1). - Vaclav Kotesovec, Oct 07 2020

A224439 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^(n-1) * x^n/n ).

Original entry on oeis.org

1, 1, 2, 7, 93, 357, 41927, 80065, 21483964, 112388242, 19973468103, 25813956365, 691174602929572, 695655501206181, 63995738768530056, 1469847380380956056, 1468171845473348201557, 1477216529008886240457, 62064992121198579569054696, 62086294811417506896412871

Offset: 0

Paul D. Hanna, Apr 06 2013

nonn

Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ), where sigma(n) = A000203(n) is the sum of the divisors of n.

			G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 93*x^4 + 357*x^5 + 41927*x^6 + ... where
log(A(x)) = x + 3^1*x^2/2 + 4^2*x^3/3 + 7^3*x^4/4 + 6^4*x^5/5 + 12^5*x^6/6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..335

Cf. A224440, A156217, A158095, A000203.

{a(n)=polcoeff(exp(sum(m=1, n, sigma(m)^(m-1)*x^m/m)+x*O(x^n)), n)}
for(n=0, 20, print1(a(n), ", "))

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

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

Logarithmic derivative yields A224440.

A158265 G.f.: A(x) = exp( Sum_{n>=1} 2sigma(n,n+1)x^n/n ).

Original entry on oeis.org

1, 2, 11, 74, 697, 8002, 115158, 1949640, 38662510, 872245634, 22150393253, 623661939852, 19296665400632, 650198159192554, 23700604926216759, 928939297013294294, 38956230043045053042, 1740248411222193973416

Offset: 0

Paul D. Hanna, Mar 29 2009

nonn

Definition: sigma(n,n+1) = Sum_{d|n} d^(n+1): [1,9,82,1057,15626,...].

			G.f.: A(x) = 1 + 2*x + 11*x^2 + 74*x^3 + 697*x^4 + 8002*x^5 +...
log(A(x)) = 2*x + 18*x^2/2 + 164*x^3/3 + 2114*x^4/4 + 31252*x^5/5 +...

Cf. A158095, A023881, A082245.

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

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

cp's OEIS Frontend

A294645 a(n) = Sum_{d|n} d^(n+1).

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A224439 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^(n-1) * x^n/n ).

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A158265 G.f.: A(x) = exp( Sum_{n>=1} 2sigma(n,n+1)x^n/n ).

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cp's OEIS Frontend

A294645 a(n) = Sum_{d|n} d^(n+1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula

A224439 G.f.: A(x) = exp( Sum_{n>=1} sigma(n)^(n-1) * x^n/n ).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A158265 G.f.: A(x) = exp( Sum_{n>=1} 2*sigma(n,n+1)*x^n/n ).

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Author

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Comments

Examples

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Programs

PARI

Formula

A158265 G.f.: A(x) = exp( Sum_{n>=1} 2sigma(n,n+1)x^n/n ).