A328311 a(n) = 1 + A051903(A003415(n)) - A051903(n), a(1) = 0 by convention.

0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 3, 0, 2, 3, 2, 0, 0, 0, 2, 1, 1, 0, 0, 0, 1, 1, 4, 0, 1, 0, 0, 1, 1, 2, 1, 0, 1, 4, 0, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 2, 2, 0, 2, 4, 0, 1, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 2, 1, 1, 0, 0, 0, 1, 0, 3, 2, 1, 0, 1, 0, 1, 0, 1, 1, 2, 5, 0, 0, 0, 2, 4, 1, 2, 3, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1

Offset: 1

Author

Antti Karttunen, Oct 13 2019

Keywords

• nonn

Comments

All terms are nonnegative because taking the arithmetic derivative (A003415) of n may decrease its "degree" (i.e., its maximal exponent, A051903) by at most one, and in many cases may also increase it, or keep it same.

Links

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

Crossrefs

One more than A328310.

Cf. A003415, A051903, A328312.

Cf. A328320 (indices of zeros), A328321 (of nonzero terms).

Programs

PARI

A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
A051903(n) = if((1==n),0,vecmax(factor(n)[, 2]));
A328311(n) = if(n<=1,0,1+(A051903(A003415(n)) - A051903(n)));

Formula

a(1) = 0, for n > 1, a(n) = 1 + A051903(A003415(n)) - A051903(n).

For n > 1, a(n) = 1 + A328310(n).