A324179 Number of distinct values A297167 obtains over divisors > 1 of n, minus number of prime factors of n counted with multiplicity: a(n) = A324190(n) - A001222(n).

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, 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, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 2, 0

Offset

1,56

Comments

a(n) is zero for all prime powers (A000961), but also for many other numbers.

Links

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

Index entries for sequences computed from indices in prime factorization

Formula

a(n) = A324190(n) - A001222(n).

a(n) <= A324192(n).

Example

Divisors of 56 larger than 1 are [2, 4, 7, 8, 14, 28, 56]. When A297167 is applied to each, one obtains values: [0, 1, 3, 2, 3, 4, 5], of which 6 values are distinct (as one of them, 3, occurs twice). On the other hand, 56 = 2 * 2 * 2 * 7 has four prime factors in total, thus a(56) = 6 - 4 = 2.

Prog

(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))));
A324179(n) = (A324190(n)-bigomega(n));

Crossrefs

Cf. A000961, A001222, A061395, A297112, A297167, A324120, A324190, A324191, A324192.

Sequence in context: A089807 A089810 A214411 * A216577 A096562 A096563

Adjacent sequences: A324176 A324177 A324178 * A324180 A324181 A324182

Keyword

nonn

Author

Antti Karttunen, Feb 19 2019

Status

approved