A327932 a(n) = A327928(n) - A129251(n), where A327928(n) gives the number of distinct primes p such that p^p divides the arithmetic derivative of n, and A129251(n) gives the number of such primes for n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Author

Antti Karttunen, Oct 01 2019

Keywords

• nonn

Examples

			For n = 15 = 3*5, A129251(15) = 0, but for A003415(15) = 8 = 2^3, A129251(8) = 1, thus a(15) = 1.
For n = 515 = 5*103, A129251(515) = 0, but for A003415(515) = 108 = 2^2 * 3^3, A129251(108) = 2, thus a(515) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A003415, A129251, A327928, A327934.

Programs

PARI

A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
A129251(n) = { my(f = factor(n)); sum(k=1, #f~, (f[k, 2]>=f[k, 1])); };
A327928(n) = if(n<=1,0,A129251(A003415(n)));
A327932(n) = (A327928(n)-A129251(n));

Formula

a(n) = A327928(n) - A129251(n).

For n > 1, a(n) = A129251(A003415(n)) - A129251(n).