A324190 Number of distinct values A297167 obtains over the divisors > 1 of n; a(1) = 0.

0, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 3, 1, 2, 2, 4, 1, 3, 1, 4, 2, 2, 1, 4, 2, 2, 3, 4, 1, 3, 1, 5, 2, 2, 2, 4, 1, 2, 2, 5, 1, 3, 1, 4, 3, 2, 1, 5, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 4, 1, 2, 4, 6, 2, 3, 1, 4, 2, 3, 1, 5, 1, 2, 3, 4, 2, 3, 1, 6, 4, 2, 1, 5, 2, 2, 2, 6, 1, 4, 2, 4, 2, 2, 2, 6, 1, 3, 4, 5, 1, 3, 1, 6, 3

Offset: 1

Author

Antti Karttunen, Feb 19 2019

Keywords

• nonn

Comments

Number of distinct values of the sum {excess of d} + {the index of the largest prime factor of d} (that is, A046660(d) + A061395(d)) that occurs over all divisors d > 1 of n.

Number of distinct values A297112 obtains over the divisors > 1 of n; a(1) = 0.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences computed from indices in prime factorization

Crossrefs

Cf. A000005, A001221, A001222, A046660, A061395, A324120, A324179, A324191, A324192, A324202, A324203.

Cf. also A156552, A297112.

Programs

PARI

A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
A324190(n) = #Set(apply(A297167, select(d -> d>1,divisors(n))));

Formula

a(n) = A001221(A324202(n)).

a(n) >= A324120(n).

a(n) >= A001222(n) >= A001221(n). [See A324179 and A324192 for differences]

a(n) <= A000005(n)-1. [See A324191 for differences]

For all primes p, a(p^k) = k.