A192015 - OEIS

%I #24 Aug 16 2024 06:09:50

%S 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,

%T 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,

%U 1,1,1,1,1,1,1,405,1,1024,1,1,1,1,1,1,1,34

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

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

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

%C A001787 and A024622 give record values and where they occur;

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

%H Reinhard Zumkeller, <a href="/A192015/b192015.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePower.html">Prime Power</a>

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

%t 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 *)

%o (Haskell)

%o a192015 = a003415 . a000961  -- _Reinhard Zumkeller_, Apr 16 2014

%o (Python)

%o from sympy import primepi, integer_nthroot, factorint

%o def A192015(n):

%o     if n == 1: return 0

%o     def f(x): return int(n+x-1-sum(primepi(integer_nthroot(x,k)[0]) for k in range(1,x.bit_length())))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return sum((m*e//p for p,e in factorint(m).items())) # _Chai Wah Wu_, Aug 15 2024

%K nonn

%O 1,4

%A _Reinhard Zumkeller_, Jun 26 2011