A099306 n''', the third arithmetic derivative of n.

0, 0, 0, 0, 4, 0, 0, 0, 32, 1, 0, 0, 80, 0, 5, 16, 176, 0, 7, 0, 48, 1, 0, 0, 112, 1, 12, 27, 176, 0, 0, 0, 368, 6, 0, 32, 96, 0, 7, 80, 156, 0, 0, 0, 240, 32, 7, 0, 608, 6, 16, 44, 96, 0, 216, 80, 272, 1, 0, 0, 272, 0, 9, 24, 2368, 10, 0, 0, 220, 8, 0, 0, 284, 0

Offset: 0

Author

T. D. Noe, Oct 12 2004

Keywords

• nonn

Comments

For prime p, a(p^p) = p^p.

a(A157037(n)) = 0. - Reinhard Zumkeller, Feb 22 2009

References

• See A003415.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 2000 terms from T. D. Noe)

Crossrefs

Cf. A003415 (arithmetic derivative of n), A068346 (second arithmetic derivative of n), A099307 (least k such that the k-th arithmetic derivative of n is zero).

Column k=3 of A258651.

Programs

Mathematica

dn[0]=0; dn[1]=0; dn[n_]:=Module[{f=Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus@@(n*f[[2]]/f[[1]])]]; Table[dn[dn[dn[n]]], {n, 100}]

Formula

a(n) = A003415(A003415(A003415(n))).