A348513 Möbius transform of A048146, the sum of non-unitary divisors of n.

0, 0, 0, 2, 0, 0, 0, 4, 3, 0, 0, 6, 0, 0, 0, 8, 0, 6, 0, 10, 0, 0, 0, 12, 5, 0, 9, 14, 0, 0, 0, 16, 0, 0, 0, 24, 0, 0, 0, 20, 0, 0, 0, 22, 15, 0, 0, 24, 7, 10, 0, 26, 0, 18, 0, 28, 0, 0, 0, 30, 0, 0, 21, 32, 0, 0, 0, 34, 0, 0, 0, 48, 0, 0, 15, 38, 0, 0, 0, 40, 27, 0, 0, 42, 0, 0, 0, 44, 0, 30, 0, 46, 0, 0, 0, 48, 0, 14

Offset: 1

Author

Antti Karttunen, Oct 29 2021

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A008683, A048146, A254503.

Programs

Mathematica

nusigma[1] = 0; nusigma[n_] := DivisorSigma[1, n] - Times @@ (1 + Power @@@ FactorInteger[n]); a[n_] := DivisorSum[n, MoebiusMu[n/#]*nusigma[#] &]; Array[a, 100] (* Amiram Eldar, Oct 30 2021 *)

PARI

A254503(n) = {my(f = factor(n)); for (i=1, #f~, if ((e=f[i, 2]) > 1, f[i, 1] = eulerphi(f[i, 1]^e); f[i, 2] = 1); ); factorback(f); }; \\ From A254503
A348513(n) = (n - A254503(n));

Formula

a(n) = n - A254503(n).

a(n) = Sum_{d|n} (A008683(n/d) * A048146(d)).