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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.

See A292867 for comments, linked tables, and conjectures. - Michael De Vlieger, Nov 17 2017

Consider A243823(n), which is the number of m < n that are products of at least one prime p | n and at least one prime q that does not divide n. These numbers m in the cototient of n do not divide a power of n. This sequence lists numbers n where such numbers m are predominant.

Odd terms in A294575.

2^k - 1 is in the sequence for k = 12, 20, 24, 28, 30, 36, 40, 48, ...

Observations:

1. There is a large gap between a(19) and a(20).

2. Products 2^5 * prime(i), with 3 <= i <= 17, are in the sequence.

3. Products 2^6 * prime(j), with 43391 <= j <= 82025, are in the sequence.

4. a(1) = 2^2 * 3 * 13, and terms 190, 286, and 578 are even, but do not follow the pattern of 2^h*p prime.

Consider numbers m that are nondivisors in the cototient of n, listed in row n of A133995 and counted by A045763(n). This sequence lists numbers n for which there are more m such that m | n^e with e >= 0 than there are m that are products of at least one prime divisor p of n and one nondivisor prime q. The former species of m are "semidivisors" listed in row n of A272618 and counted by A243822(n), while the latter are "semitotatives" listed in row n of A272619 and counted by A243823(n). These two species constitute the only species of nondivisors in the cototient of n.

Primes p have no nondivisors in the cototient, i.e., A045763(p) = 0, therefore A243822(p) and A243823(p) also are 0. The equality of these latter two sequences is trivial in the case of primes.

Prime powers p^e except for p^e = 4 have A243823(p^e) > A243822(p^e), since A243822(p^e) = 0. All powers p^k with 0 <= k <= e divide p^e.

The sequence is finite because there exist a lot more nondivisor primes q than p | n as n increases. Therefore there are more numbers m in row n of A272619 than there are in row n of A272618, since the former are products p*q and the latter are products only of p.

Former name: number of "semidivisors" of n, numbers m < n that do not divide n but divide n^e for some integer e > 1. See ACM Inroads paper.

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.

The k are the "semitotatives" of n as counted by A243823(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. Further, the terms k must have at least two distinct prime divisors p and q.

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 all the numbers k in the corresponding row of A133995. There is only one prime divisor p of p^e and every power 1 <= m <= e of p divides p^e, thus none of the terms of the corresponding row of A133995 are in A272618(n).

Rows n = 4 and 6 are special cases of composite n that contains zero. 4 is the smallest composite number; there are no composites k < n. 6 has the prime divisors 2 and 3, thus 5 is the smallest prime coprime to 6; the product of the minimum prime divisor and minimum prime coprime to 6 is 10, which exceeds 6 and falls outside the considered range. The situation is not so for composite n > 6. Thus rows n for composite n > 6 contain at least 1 nonzero value.

The smallest k of row n = A096014(n) < n, i.e., those values of A096014(n) pertaining to composite n > 6, a product of the smallest prime divisor p of n and the smallest prime q coprime to n. The smallest k of n are even squarefree semiprimes since 2 either divides n or is coprime to n and k is by definition a number with at least two distinct primes. The smallest k = 2p for p^2 sets record values for A096014(n) when we ignore values pertaining to prime n, n = 4, and n = 6.

In base n, 1/a(n) has a mixed recurrent expansion.