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

0,82

Links

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

Formula

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

Example

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.

Prog

(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)));

Crossrefs

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

Sequence in context: A236233 A112607 A161371 * A364387 A147645 A091970

Adjacent sequences: A327925 A327926 A327927 * A327929 A327930 A327931

Keyword

nonn

Author

Antti Karttunen, Oct 01 2019

Status

approved