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We define an "n-regular" number as 1 <= m <= n such that m | n^e with integer e >= 0. The divisor d is a special case of regular number m such that d | n^e with e = 0 or e = 1. Regular numbers m can exceed n; we are concerned only with regulars m <= n herein.

Since divisors are a special case of regular numbers, we examine those numbers m that set records for the ratio of the "regular counting function" A010846(m) and the divisor counting function A000005(m).

There are 2 nonsquarefree terms {18, 150} less than 36,000,000.

The sequence contains no numbers with omega(m) = 1. This is because all regular m divide p^e, and since all the regulars of 1 also divide 1, no primes or prime powers greater than 1 appear in a(n).

The values of A000005(a(n)) are in A000079, i.e., powers 2^e except e = 1.

Aside from the 2 nonsquarefree terms, many terms m are products of A002110(i) * p_j, with j > i between some lower and upper bound outside of when m is in A002110. Example: 30 is in A002110; {42, 66, 78, 102, 114, 138, 174} are A002110(3) * p_j with 2 <= j <= 8.

There are a few terms of the form A002110(i) * p_j * p_k, with i + 1 < j < k. In other words, there is a gap in the indices of the prime divisors between the 3rd and 2nd largest prime divisors, as well as one potentially between the 2nd and largest prime divisors. The smallest m of this type is 46410 = 2 * 3 * 5 * 7 * 13 * 17, followed by 51870 = 2 * 3 * 5 * 7 * 13 * 19.

Conjectures:

1. The only nonsquarefree terms are 18 and 150.

2. Primorials A002110(i) for i = 0 and i > 2 are in the sequence.

Let R = r(P(n)) = A010846(A002110(n)) = A363061(n).

Let S(n) be the sorted tensor product of prime power ranges {p(i)^e : i<=n, e>=0}, e.g., S(1) = A000079, S(2) = A003586, S(3) = A051037, etc.

Let T(n) = A002110(n)*S(n). Note that S(1) = T(1) since omega(A002110(1)) = 1.

Let S(n,i) be the i-th term in S(n).

Then a(n) is the smallest S(n,i), i >= R, such that S(n,i) is also in T. Equivalently, a(n) is the smallest S(n,i), i >= R, such that rad(S(n,i)) = A002110(n), where rad(n) = A007947(n).

Row n begins with 1, ends with n, and has A010846(n) terms.

From Michael De Vlieger, Jul 08 2014: (Start)

Prime p has {1, p} and A010846(p) = 2.

Prime power p^e has {1, p, ..., p^e} and A010846(p^e) = A000005(p^e) = e + 1.

Composite c that are not prime powers have A010846(c) = A000005(c) + A243822(c), where A243822(c) is nonzero positive, since the minimum prime divisor p of c produces at least one semidivisor (e.g., p^2 < c). Thus these have the set of divisors of c and at least one semidivisor p^2. For squareful c that are not prime powers, p^2 may divide c, but p^3 does not. The minimum squareful c = 12, 2^3 does not divide 12 yet is less than 12 and is a product of the minimum prime divisor of 12. All other even squareful c admit a power of 2 that does not divide c, since there must be another prime divisor q > 2. (End)

Numbers 1 <= k <= n such that (floor(n^k/k) - floor((n^k - 1)/k)) = 1. - Michael De Vlieger, May 26 2016

Numbers 1 <= k <= n such that k | n^e with e >= 0. - Michael De Vlieger, May 29 2018

For n > 1, a(n) gives the (one-based) index of the row where n is located in arrays A284311 and A285321 or respectively, index of the column where n is in A284457. A285329 gives the other index. - Antti Karttunen, Apr 17 2017

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.

Semitotatives m < n have a regular factor that is the product of prime divisors of n, and a coprime factor that is the product of primes q that are coprime to n.

The unit fractions of semitotatives have a mixed recurrent expansion in base n (See Hardy & Wright).

a(n) = 0 for prime powers and squarefree numbers.

Differs from A284318 after 27 terms.

Let rad(x) = A007947(x).

Let T(n,k) be the k-th term of row n in this sequence.

Define S(n,k) to be the k-th term in row n of A162306.

T(n,k) = rad(n) * S(n,k), k <= A008479(n).

The number n appears as the last term in row n.

This sequence is the product of n-regular numbers.

A number m is said to be "regular" to n or "n-regular" if all the prime factors p of m also divide n.

The divisor is a special case of a regular m such that m also divides n in addition to all of its prime factors p | n.

Analogous to A007955 (Product of divisors of n).

If n is 1 or prime, a(n) = n.

If n is a prime power, a(n) = A007955(n).

Note: b-file ends at n = 4619, because a(4620) has more than 1000 decimal digits.

Product of the numbers 1 <= k <= n such that (floor(n^k/k) - floor((n^k - 1)/k)) = 1. - Michael De Vlieger, May 26 2016