A383783 a(n) = Sum_{k=1..2^n} mu(k) * (floor(2^n/k)^4 - floor((2^n-1)/k)^4).

1, 14, 160, 1520, 13216, 110144, 899200, 7266560, 58425856, 468583424, 3753379840, 30045900800, 240442679296, 1923843375104, 15391954862080, 123140470538240, 985143091265536, 7881222038749184, 63050085546065920, 504401921315962880, 4035220318323736576

Offset

0,2

Links

Chai Wah Wu, Table of n, a(n) for n = 0..48

Formula

a(n) = A344597(2^n) = A082540(2^n) - A082540(2^n-1).

Mathematica

a[n_]:=Sum[MoebiusMu[k]*(Floor[2^n/k]^4-Floor[(2^n-1)/k]^4), {k, 2^n}]; Array[a, 21, 0] (* James C. McMahon, May 10 2025 *)

Prog

(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 A383783(n): return A082540(m:=1<<n)-A082540(m-1)
(PARI) a(n) = sum(k=1, 2^n, moebius(k) * ((2^n\k)^4 - ((2^n-1)\k)^4)); \\ Michel Marcus, May 10 2025

Crossrefs

Cf. A082540, A344597.

Sequence in context: A268946 A383946 A343093 * A229611 A282043 A193103

Adjacent sequences: A383780 A383781 A383782 * A383784 A383785 A383786

Keyword

nonn

Author

Chai Wah Wu, May 09 2025

Status

approved