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A295127 Numerator of Sum_{d|n} mu(n/d)/d, where mu is the Möbius function A008683.

%I A295127 #32 Jun 06 2025 08:34:21
%S A295127 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,
%T A295127 6,-2,3,-28,-4,-30,-1,20,8,24,1,-36,9,8,1,-40,-2,-42,5,8,11,-46,1,-6,
%U A295127 2,32,3,-52,1,8,3,12,14,-58,-2,-60,15,4,-1,48,-10,-66,4,44,-12
%N A295127 Numerator of Sum_{d|n} mu(n/d)/d, where mu is the Möbius function A008683.
%C A295127 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, ...;
%C A295127 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, ..., .
%H A295127 Mats Granvik and Robert G. Wilson v, <a href="/A295127/b295127.txt">Table of n, a(n) for n = 1..10000</a>
%F A295127 From _Amiram Eldar_, Jun 06 2025: (Start)
%F A295127 f(n) = a(n)/A295126(n) is multiplicative with f(p^e) = -(p-1)/p^e.
%F A295127 Dirichlet g.f. of f(n): zeta(s+1)/zeta(s). (End)
%e A295127 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.
%p A295127 f:= n -> numer(add(numtheory:-mobius(n/k)/k, k=numtheory:-divisors(n))):
%p A295127 map(f, [$1..100]); # _Robert Israel_, Nov 16 2017
%t A295127 f[n_] := Block[{d = Divisors@ n}, Plus @@ (MoebiusMu[d]/Reverse@ d)]; Numerator@ Array[f, 70]
%t A295127 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 *)
%o A295127 (PARI) a(n) = numerator(sumdiv(n, d, moebius(n/d)/d)); \\ _Michel Marcus_, Nov 17 2017
%Y A295127 Cf. A008683, A191898, A007913, A023900, A173557, A295126 (denominator).
%K A295127 sign,look,frac,easy
%O A295127 1,3
%A A295127 _Mats Granvik_ and _Robert G. Wilson v_, Nov 15 2017