A348976 Möbius transform of A129283, which is sum of n and its arithmetic derivative.

1, 2, 3, 5, 5, 5, 7, 12, 11, 9, 11, 12, 13, 13, 14, 28, 17, 17, 19, 22, 20, 21, 23, 28, 29, 25, 39, 32, 29, 22, 31, 64, 32, 33, 34, 40, 37, 37, 38, 52, 41, 32, 43, 52, 50, 45, 47, 64, 55, 49, 50, 62, 53, 57, 54, 76, 56, 57, 59, 52, 61, 61, 72, 144, 64, 52, 67, 82, 68, 58, 71, 92, 73, 73, 78, 92, 76, 62, 79, 120

Offset: 1

Author

Antti Karttunen, Nov 09 2021

Keywords

• nonn

Links

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

Crossrefs

Cf. A000010, A003415, A008683, A129283, A300251, A347084, A348970.

Programs

Mathematica

f[p_, e_] := e/p; d[1] = 1; d[n_] := n*(1 + Plus @@ f @@@ FactorInteger[n]); a[n_] := DivisorSum[n, MoebiusMu[#]*d[n/#] &]; Array[a, 100] (* Amiram Eldar, Nov 13 2021 *)

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A129283(n) = (n+A003415(n));
A348976(n) = sumdiv(n,d,moebius(n/d)*A129283(d));

Formula

a(n) = Sum_{d|n} A008683(n/d) * A129283(d).

a(n) = A000010(n) + A300251(n).