A082878 Arithmetic derivative of numbers of the form p*q^k with distinct primes p and q, k > 0.

5, 7, 16, 9, 8, 21, 24, 10, 13, 44, 15, 32, 14, 19, 12, 21, 16, 68, 48, 39, 25, 112, 45, 20, 56, 81, 16, 92, 22, 31, 33, 51, 18, 72, 26, 39, 55, 80, 18, 176, 43, 22, 45, 32, 140, 20, 96, 34, 49, 24, 272, 77, 75, 164, 55, 40, 240, 28, 120, 87, 61, 24, 63, 44, 128, 46, 26, 69

Offset: 1

Author

Reinhard Zumkeller, May 25 2003

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003415, A084227.

Programs

Mathematica

s[n_] := Module[{f = FactorInteger[n], i}, If[Length[f] == 2 && Min[f[[;;, 2]]] == 1, i = FirstPosition[f[[;;, 2]], 1][[1]]; n * (1/f[[i, 1]] + f[[3-i, 2]] /f[[3-i, 1]]), Nothing]]; Array[s, 100] (* Amiram Eldar, Mar 25 2025 *)

Formula

a(n) = A003415(A084227(n)).

a(p*q^k) = (p+q)*q^(k-1).

Extensions

Data corrected by Amiram Eldar, Mar 25 2025