cp's OEIS Frontend

The k are the "semidivisors" or nondivisor regular numbers of n as counted by A243822(n).

All nonzero terms k are composite and pertain to composite rows n. This is because prime k must either divide or be coprime to n, and k = 1 is both a divisor of and coprime to n.

Row n for prime p contains zero, since numbers 1 <= k < p must either divide or be coprime to prime p.

Row n for prime powers p^e contains zero, since there is only one prime divisor p of p^e and every power 1 <= m <= e of p divides p^e.

Row n = 4 is a special case of composite n that contains zero. This is because 4 is the smallest composite number; there are no composites k < n.

Thus rows n for composite n > 4 contain at least 1 nonzero value.

In base n, 1/a(n) has a terminating expansion with at least 2 places.

Row n has length A264441(n).

The number of nonzero entries in row n is A045763(n).

Row n has a 0 if every positive integer <= n is coprime to n or divides n.

From Michael De Vlieger, Aug 19 2017: (Start)

When row n is not empty (and here represented by 0), the terms of row n are composite, since primes p < n must either divide or be coprime to n and the empty product 1 both divides and is coprime to all numbers. For the following, let prime p divide n and prime q be coprime to n.

Row n is empty for n < 8 except n = 6.

There are two distinct species of term m of row n. The first are nondivisor regular numbers g in A272618(n) that divide some integer power e > 1 of n. In other words, these numbers are products of primes p that also divide n and no primes q that are coprime to n, yet g itself does not divide n. Prime powers n = p^k cannot have numbers g in A272618(n) since they have only one distinct prime divisor p; all regular numbers g = p^e with 0 <= e <= k divide p^k. The smallest n = 6 for which there is a number in A272618. The number 4 is the smallest composite and is equal to n = 4 thus must divide it; 4 is coprime to 5. The number 4 is neither coprime to nor a divisor of 6.

The second are numbers h in A272619(n) that are products of at least one prime p that divides n and one prime q that is coprime to n.

The smallest n = 8 for which there is a number in A272619 is 8; the number 6 is the product of the smallest two distinct primes. 6 divides 6 and is coprime to 7. The number 6 is neither coprime to nor a divisor of the prime power 8; 4 divides 8 and does not appear in a(8).

There can be no other species since primes p <= n divide n and q < n are coprime to n, and products of primes q exclusive of any p are coprime to n.

As a consequence of these two species, rows 1 <= n <= 5 and n = 7 are empty and thus have 0 in row n.

(End)

Since A010846(n) = A000005(n) + A243822(n), this sequence examines the balance of the two components among "regular" numbers.

Value of a(n) is generally less frequently negative as n increases.

a(1) = -1.

For primes p, a(p) = -2 since 1 | p and the cototient is restricted to the divisor p.

For perfect prime powers p^e, a(p^e) = -(e + 1), since all m < p^e in the cototient of p^e that do not have a prime factor q coprime to p^e are powers p^k with 1 < p^k <= p^e; all such p^k divide p^e.

Generally for n with A001221(n) = 1, a(n) = -1 * A000005(n), since the cototient is restricted to divisors, and in the case of p^e > 4, divisors and numbers in A272619(p^e) not counted by A010846(p^e).

For m >= 3, a(A002110(m)) is positive.

For m >= 9, a(A244052(m)) is positive.

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.

Except for A292867(1) = 1, all terms are even.

Some conjectures:

1. The only prime powers p^e in this sequence are {8, 16, 32, 64}.

2. Squarefree terms m appear throughout. (There are 261 squarefree values among the first 1261 terms.)

3. Terms that set records for omega(m) are 1, followed by 2^e, with 3 <= e <= 6, then 2^e * 3 with 6 <= e <= 8, then 2^7 * A002110(k) with k >= 1.

4. Primorials A002110(n) for n >= 6 appear in this sequence. The first primorials in m are terms 6 through 8 of A002110 (i.e., 30030, 510510, 9699690) at n = 419, 774, 1258, respectively.

5. Outside of a(n) with 2 <= n <= 21 and n = {29, 30}, all terms of A244052 are also in this sequence. This observation applies to the smallest 104 terms of A244052.

6. For very large n, all terms are also in A244052. For small n, few terms of A244052 appear and are separated by many other numbers. Since numbers m in A244052 are products of k primes, many of which are the smallest primes, phi is minimized and A010846(m) becomes infinitesimal in comparison to m. Therefore A243823(m) is tantamount to the cototient of m. The size of n required to observe this agreement between this sequence and A244052 is unknown.

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.

Number of terms in row n of A272619 that are coprime to (n-1)*(n+1).

a(n) = 0 for prime powers, since the definition implies omega(n) >= 2.

a(n) is the cardinality of k in the cototient of n such that rad(k) mod rad(n) != 0 and rad(n) mod rad(k) != 0.

Analogous to A066760(n), the sum of row n of A133995, and A381499(n), sum of row n of A272619.