A192015 Arithmetic derivative of prime powers: a(n) = A003415(A000961(n)).

0, 1, 1, 4, 1, 1, 12, 6, 1, 1, 32, 1, 1, 1, 10, 27, 1, 1, 80, 1, 1, 1, 1, 14, 1, 1, 1, 192, 1, 1, 1, 1, 108, 1, 1, 1, 1, 1, 1, 1, 1, 22, 75, 1, 448, 1, 1, 1, 1, 1, 1, 1, 1, 26, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 405, 1, 1024, 1, 1, 1, 1, 1, 1, 1, 34

Offset: 1

Author

Reinhard Zumkeller, Jun 26 2011

Keywords

• nonn

Comments

a(A000040(n)) = 1; a(A002808(n)) > 1;

A001787, A027471, A100484, A079705 and A051674 are subsequences;

A001787 and A024622 give record values and where they occur;

A192016(n) = A003415(a(n)).

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Power

Programs

Haskell

a192015 = a003415 . a000961  -- Reinhard Zumkeller, Apr 16 2014

Mathematica

Join[{0}, Reap[For[n = 1, n <= 300, n++, f = FactorInteger[n]; If[Length[f] == 1, Sow[n*Total[Apply[#2/#1&, f, {1}]]]]]][[2, 1]]] (* Jean-François Alcover, Feb 21 2014 *)

Python

from sympy import primepi, integer_nthroot, factorint
def A192015(n):
    if n == 1: return 0
    def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return sum((m*e//p for p,e in factorint(m).items())) # Chai Wah Wu, Aug 15 2024

Formula

a(n) = A025474(n) * A025473(n)^(A025474(n) - 1).