A328311 - OEIS

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

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OFFSET

1,12

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

FORMULA

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

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

PROG

(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)));

CROSSREFS

One more than A328310.