A295126 Denominator of Sum_{d|n} mu(n/d)/d, where mu is the Möbius function A008683.

1, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 6, 13, 7, 15, 16, 17, 9, 19, 5, 7, 11, 23, 12, 25, 13, 27, 14, 29, 15, 31, 32, 33, 17, 35, 18, 37, 19, 13, 10, 41, 7, 43, 22, 45, 23, 47, 24, 49, 25, 51, 13, 53, 27, 11, 28, 19, 29, 59, 15, 61, 31, 21, 64, 65, 33, 67, 17, 69, 35

Offset: 1

Author

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

Keywords

• nonn
• look
• frac
• easy

Comments

a(n) <= n.

a(n) <> n when n is in A069209.

n == 0 (mod a(n)).

First occurrence of k: 1, 2, 3, 4, 5, 12, 7, 8, 9, 40, 11, 24, 13, 28, 15, 16, 17, 36, 19, 80, 63, 44, 23, 48, 25, ..., ;

First occurrence of k = a(n)/n: 1, 6, 21, 20, 55, 42, 203, 120, 171, 110, 253, 84, 689, 406, 465, 272, 1751, 342, ..., .

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, A069209, A191898, A007913, A023900, A173557, A295127 (numerator).

Programs

Maple

f:= n -> denom(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)]; Denominator@ Array[f, 70]
f[p_, e_] := -(p-1)/p^e; a[1] = 1; a[n_] := Denominator[Times @@ f @@@ FactorInteger[n]]; Array[a, 100] (* Amiram Eldar, Jun 06 2025 *)

PARI

a(n) = denominator(sumdiv(n, d, moebius(n/d)/d)); \\ Michel Marcus, Nov 17 2017