A262843 Inverse Moebius transform of n^(n-1).

1, 3, 10, 67, 626, 7788, 117650, 2097219, 43046731, 1000000628, 25937424602, 743008378540, 23298085122482, 793714773371796, 29192926025391260, 1152921504608944195, 48661191875666868482, 2185911559738739586477, 104127350297911241532842, 5242880000000001000000692, 278218429446951548637314060

Offset

1,2

Comments

Logarithmic derivative of A262842.

Links

G. C. Greubel, Table of n, a(n) for n = 1..385

Formula

a(n) = Sum{d|n} d^(d-1).

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

Example

O.g.f.: A(x) = x + 3*x^2 + 10*x^3 + 67*x^4 + 626*x^5 + 7788*x^6 +...
where
A(x) = x/(1-x) + 2*x^2/(1-x^2) + 3^2*x^3/(1-x^3) + 4^3*x^4/(1-x^4) + 5^4*x^5/(1-x^5) + 6^5*x^6/(1-x^6) +...+ n^(n-1)* x^n/(1 -x^n) +...
Logarithmic generating function.
L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 67*x^4/4 + 626*x^5/5 + 7788*x^6/6 +...
where
exp(L(x)) = 1/( (1-x) * (1-x^2) * (1-x^3)^3 * (1-x^4)^16 * (1-x^5)^125 * (1-x^6)^1296 *...* (1-x^n)^(n^(n-2)) *...).
Explicitly,
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 22*x^4 + 150*x^5 + 1469*x^6 + 18452*x^7 + 282426*x^8 +...+ A262842(n)*x^n ...

Mathematica

a[n_] := DivisorSum[n, #^(#-1) &]; Array[a, 30] (* Jean-François Alcover, Dec 23 2015 *)

Prog

(PARI) {a(n)=sumdiv(n, d, d^(d-1))}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(m=1, n, m^(m-1)*x^m/(1-x^m +x*O(x^n))), n)}
for(n=1, 30, print1(a(n), ", "))
(Python)
from sympy import divisors
def A262843(n): return sum(d**(d-1) for d in divisors(n, generator=True)) # Chai Wah Wu, Jun 19 2022

Crossrefs

Cf. A262842, A062796.

Sequence in context: A217388 A004102 A072638 * A080526 A359701 A308814

Adjacent sequences: A262840 A262841 A262842 * A262844 A262845 A262846

Keyword

nonn

Author

Paul D. Hanna, Oct 03 2015

Status

approved