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

Author

Mats Granvik and Robert G. Wilson v, Nov 15 2017

Keywords

• sign
• look
• frac
• easy

Comments

First occurrence of k > 0: 1, 10, 14, 21, 22, 26, 116, 15, 38, 164, 46, 57, 212, 58, 62, 390, 1096, 74, 45840, 33, ...;

First occurrence of k < 0: 2, 3, 228, 5, 132, 7, 516, 204, 728, 11, 276, 13, 948, 258, 308, 17, 1236, 19, 5496, 366, ..., .

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

Crossrefs

Cf. A008683, A191898, A007913, A023900, A173557, A295126 (denominator).

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)