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

1,515

Links

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

Formula

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

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

Example

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.

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)));
A327932(n) = (A327928(n)-A129251(n));

Crossrefs

Cf. A003415, A129251, A327928, A327934.

Sequence in context: A353479 A360111 A359162 * A359546 A354989 A277161

Adjacent sequences: A327929 A327930 A327931 * A327933 A327934 A327935

Keyword

nonn

Author

Antti Karttunen, Oct 01 2019

Status

approved