A192134 Difference between n-th prime power and its arithmetic derivative.

1, 1, 2, 0, 4, 6, -4, 3, 10, 12, -16, 16, 18, 22, 15, 0, 28, 30, -48, 36, 40, 42, 46, 35, 52, 58, 60, -128, 66, 70, 72, 78, -27, 82, 88, 96, 100, 102, 106, 108, 112, 99, 50, 126, -320, 130, 136, 138, 148, 150, 156, 162, 166, 143, 172, 178, 180, 190, 192, 196

Offset: 1

Author

Reinhard Zumkeller, Jun 26 2011

Keywords

• sign
• easy

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000961, A001248, A003415, A005722, A006093, A025473, A025474, A095874, A192015, A192133.

Programs

Haskell

a192134 n = a000961 n - a192015 n  -- Reinhard Zumkeller, Apr 16 2014

Mathematica

f[n_] := If[n == 1, 1, If[PrimePowerQ[n], {p, e} = FactorInteger[n][[1]]; n - e*p^(e-1), Nothing]]; Array[f, 300] (* Amiram Eldar, Apr 11 2025 *)

Formula

a(n) = A000961(n)-A192015(n) = A000961(n)-A003415(A000961(n)) = A192133(n)*A025473(n)^(A025474(n)-1) = A192133(n)*A000961(n)/A025473(n).

a(A095874(A000040(n))) = A006093(n).

a(A095874(A001248(n))) = A005722(n) + 1.