cp's OEIS Frontend

The cototient of n consists of numbers 1 < m <= n that are not coprime to n, i.e., gcd(m,n) > 1. These numbers have at least one prime divisor p that also divides n. The cototient of n contains the divisors d of n; the remaining nondivisors in the cototient of n are listed in A133995. The counting function of A133995 is A045763(n). There are two species of numbers in the nondivisor-cototient of n: those in row n of A272618, of which A243822(n) is counting function, and those in row n of A272619, of which A243823(n) is the counting function. The former species divides n^e for integer e > 1, while the latter does not divide any integer power of n.

A045763(p) = 0 for p prime, therefore there are no primes in a(n).

Except for prime terms (i.e., 2), A002110 is a subset as primorials minimize the totient function. The divisor counting function is increasingly vanishingly small compared to the totient function for A002110(i) as i increases, and A002110(i) for 1 < i <= 9 is observed in a(n).

Conjectures based on 1255 terms of a(n) < 36,000,000:

1. There are no prime powers p^e > 1 in a(n), i.e., the intersection of a(n) and A000961 is {1}.

2. A293555 is a subset of A300859. Numbers that have a lot of nondivisors m | n^e with e > 1 (i.e., in row n of A272618 and counted by A243822(n)) tend to reduce the totient and increasingly have fewer divisors than highly composite numbers, widening the nondivisor-cototient.

3. A300156 is a subset of A300859. Numbers that have more nondivisors m | n^e with e > 1 (i.e., in row n of A272618 and counted by A243822(n)) than divisors tend to reduce the totient and have fewer divisors than highly composite numbers (i.e., those n in A002182), widening the nondivisor-cototient.

Increasingly many terms k in A262867 also appear in a(n) as k increases. A292867 lists record-setters in A243823, which is the counting function of one of the two species of nondivisors in the cototient of n.

See A293555 for comments, linked tables, and conjecture. - Michael De Vlieger, Nov 17 2017