A344597 a(n) = Sum_{k=1..n} mu(k) * (floor(n/k)^4 - floor((n-1)/k)^4).

1, 14, 64, 160, 368, 592, 1104, 1520, 2400, 3056, 4640, 5264, 7824, 8736, 11776, 13216, 17984, 18384, 25344, 26080, 33312, 35120, 45584, 44320, 58480, 58512, 72000, 73200, 92624, 86848, 113520, 110144, 132640, 132416, 162816, 152112, 194544, 185616, 220416

Offset: 1

Author

Seiichi Manyama, May 24 2021

Keywords

• nonn

Crossrefs

Cf. A000583, A082540, A140434, A344596.

Programs

Mathematica

a[n_] := Sum[MoebiusMu[k] * First @ Differences @ (Quotient[{n - 1, n}, k]^4), {k, 1, n}]; Array[a, 50] (* Amiram Eldar, May 24 2021 *)

PARI

a(n) = sum(k=1, n, moebius(k)*((n\k)^4-((n-1)\k)^4));

PARI

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, moebius(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^4))

Python

from functools import lru_cache
@lru_cache(maxsize=None)
def A082540(n):
    if n == 0:
        return 0
    c, j = 1, 2
    k1 = n//j
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A082540(k1)
        j, k1 = j2, n//j2
    return n*(n**3-1)-c+j
def A344597(n): return A082540(n)-A082540(n-1) # Chai Wah Wu, May 09 2025

Formula

Sum_{k=1..n} a(k) * floor(n/k) = n^4.

Sum_{k=1..n} a(k) = A082540(n).

G.f.: Sum_{k >= 1} mu(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^4.