A295127 Numerator of Sum_{d|n} mu(n/d)/d, where mu is the Möbius function A008683.
1, -1, -2, -1, -4, 1, -6, -1, -2, 2, -10, 1, -12, 3, 8, -1, -16, 1, -18, 1, 4, 5, -22, 1, -4, 6, -2, 3, -28, -4, -30, -1, 20, 8, 24, 1, -36, 9, 8, 1, -40, -2, -42, 5, 8, 11, -46, 1, -6, 2, 32, 3, -52, 1, 8, 3, 12, 14, -58, -2, -60, 15, 4, -1, 48, -10, -66, 4, 44, -12
Offset: 1
Examples
a(6) = 1 since mu(6)/1 + mu(3)/2 + mu(2)/3 + mu(1)/6 = 1 - 1/2 - 1/3 + 1/6 = 1/3.
Links
- Mats Granvik and Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Programs
-
Maple
f:= n -> numer(add(numtheory:-mobius(n/k)/k, k=numtheory:-divisors(n))): map(f, [$1..100]); # Robert Israel, Nov 16 2017
-
Mathematica
f[n_] := Block[{d = Divisors@ n}, Plus @@ (MoebiusMu[d]/Reverse@ d)]; Numerator@ Array[f, 70] f[p_, e_] := -(p-1)/p^e; a[1] = 1; a[n_] := Numerator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 06 2025 *) -
PARI
a(n) = numerator(sumdiv(n, d, moebius(n/d)/d)); \\ Michel Marcus, Nov 17 2017
Formula
From Amiram Eldar, Jun 06 2025: (Start)
f(n) = a(n)/A295126(n) is multiplicative with f(p^e) = -(p-1)/p^e.
Dirichlet g.f. of f(n): zeta(s+1)/zeta(s). (End)
Comments