A258652 Sum of the k-th arithmetic derivative of n-k for k=0..n.

0, 1, 2, 4, 5, 9, 11, 16, 14, 25, 36, 59, 99, 209, 419, 860, 1730, 3862, 9464, 21868, 74371, 244648, 727345, 3098351, 13469007, 56269849, 281642632, 1406177909, 9597415332, 58891421656, 411673964638, 3406742649805, 24202753250241, 176482943622608

Offset: 0

Author

Alois P. Heinz, Jun 06 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Wikipedia, Arithmetic derivative

Crossrefs

Antidiagonal sums of A258651.

Cf. A003415.

Programs

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
A:= proc(n, k) option remember; `if`(k=0, n, d(A(n, k-1))) end:
a:= proc(n) option remember; add(A(h, n-h), h=0..n) end:
seq(a(n), n=0..40);

Mathematica

d[n_ /; n>1] := n*Sum[i[[2]]/i[[1]], {i, FactorInteger[n]}]; d[_] = 0;
A[n_, k_] := A[n, k] = If[k == 0, n, d[A[n, k-1]]];
a[n_] := a[n] = Sum[A[h, n-h], {h, 0, n}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Jun 01 2018, from Maple *)

Formula

a(n) = Sum_{k=0..n} A258651(n-k,k).