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

1,4

Comments

Inverse Möbius transform of A003415.

Links

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

Formula

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

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

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 *)

Prog

(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));

Crossrefs

Cf. A003415, A319683.

Cf. also A300251, A300252, A305809, A317835.

Sequence in context: A089027 A354432 A023890 * A102778 A135544 A051724

Adjacent sequences: A319681 A319682 A319683 * A319685 A319686 A319687

Keyword

nonn

Author

Antti Karttunen, Oct 02 2018

Status

approved