A344429 a(n) = Sum_{k=1..n} mu(k) * k^n.

1, -3, -34, -96, -3399, 30239, -624046, -4482626, -32249230, 9768165230, -186975207617, -2150337557747, -327482869358214, 6894274639051756, 539094536846680025, 8044964790023844733, -707278869236116107432, -12275330572755863672628, -2190860499375418948848067

Offset: 1

Author

Seiichi Manyama, May 19 2021

Keywords

• sign

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..386

Crossrefs

Cf. A002321, A008683, A031971, A068340, A321222, A332468, A336276, A336277, A336278, A336279, A344430.

Programs

Mathematica

a[n_] := Sum[MoebiusMu[k] * k^n, {k,1,n}]; Array[a, 20] (* Amiram Eldar, May 19 2021 *)

PARI

a(n) = sum(k=1, n, moebius(k)*k^n);

Python

from functools import lru_cache
from math import comb
from sympy import bernoulli
@lru_cache(maxsize=None)
def faulhaber(n,p):
    """ Faulhaber's formula for calculating Sum_{k=1..n} k^p
        requires sympy version 1.12+ where bernoulli(1) = 1/2
    """
    return sum(comb(p+1,k)*bernoulli(k)*n**(p-k+1) for k in range(p+1))//(p+1)
@lru_cache(maxsize=None)
def A344429(n,m=None):
    if n <= 1:
        return 1
    if m is None:
        m=n
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (faulhaber(j-1,m)-faulhaber(j2-1,m))*A344429(k1,m)
        j, k1 = j2, n//j2
    return c+faulhaber(j-1,m)-faulhaber(n,m) # Chai Wah Wu, Nov 02 2023