A319684 Sum of A003415(d) over the divisors d of n, where A003415 is arithmetic derivative.

0, 1, 1, 5, 1, 7, 1, 17, 7, 9, 1, 27, 1, 11, 10, 49, 1, 34, 1, 37, 12, 15, 1, 83, 11, 17, 34, 47, 1, 54, 1, 129, 16, 21, 14, 114, 1, 23, 18, 117, 1, 68, 1, 67, 55, 27, 1, 227, 15, 64, 22, 77, 1, 142, 18, 151, 24, 33, 1, 190, 1, 35, 69, 321, 20, 96, 1, 97, 28, 90, 1, 326, 1, 41, 75, 107, 20, 110, 1, 325, 142, 45, 1, 244, 24, 47, 34, 219, 1, 243, 22

Offset: 1

Author

Antti Karttunen, Oct 02 2018

Keywords

• nonn

Comments

Inverse Möbius transform of A003415.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003415, A319683.

Cf. also A300251, A300252, A305809, A317835.

Programs

Mathematica

Block[{a}, a[1] = 0; a[n_] := a[n] = If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Array[DivisorSum[#, a[#] &] &, 91]] (* Michael De Vlieger, May 24 2021 *)

PARI

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A319684(n) = sumdiv(n,d,A003415(d));

Formula

a(n) = Sum_{d|n} A003415(d).

a(n) = A319683(n) + A003415(n).