cp's OEIS Frontend

Consider numbers in the cototient of n, listed in row n of A121998. For composite n > 4, there are nondivisors m in the cototient, listed in row n of A133995. Of these m, there are two species. The first are m that divide n^e with integer e > 1, while the last do not divide n^e. These are listed in row n of A272618 and A272619, and counted by A243822(n) and A243823(n), respectively. This sequence is the difference between the latter and the former species of nondivisors in the cototient of n.

Since A045763(n) = A243822(n) + A243823(n), this sequence examines the balance of the two components among nondivisors in the cototient of n.

For positive n < 6 and for p prime, a(n) = a(p) = 0, thus a(A046022(n)) = 0.

For prime powers p^e, a(p^e) = A243823(p^e), since A243822(p^e) = 0, thus a(n) = A243823(n) for n in A000961.

Value of a(n) is generally nonnegative; a(n) is negative for n = {6, 10, 12, 18, 30}; a(30) = -5, but a(n) = -1 for the rest of the aforementioned numbers. These five numbers are a subset of A295523.

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.