A327928 Number of distinct primes p such that p^p divides the arithmetic derivative of n.

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 0

Author

Antti Karttunen, Oct 01 2019

Keywords

• nonn

Examples

			For n=20, A003415(20) = 24 = 2^3 * 3^1, thus only 2^2 divides 24, and a(24) = 1.
For n=81, A003415(81) = 108 = 2^2 * 3^3. Both 2^2 and 3^3 divide 108, thus a(81) = 2.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65537

Crossrefs

Cf. A003415, A129251, A327929 (indices of nonzero terms), A327932.

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)));

Formula

a(0) = a(1) = 0; for n > 1, a(n) = A129251(A003415(n)).