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Analogous to highly divisible numbers (A002182).

Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.

Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.

a(n) is the length of row A372111(n) of A162306.

Analogous to A371909, which instead regards A109890 and A109735.

These numbers k have the smallest A010846(k) for a number with n distinct prime factors.

a(7) <= 1398483454696343742813089 = 1049 * 2819 * 3319 * 3433 * 3457 * 3463 * 3467.

a(8) <= 32829974457045619959776094471833047127947.

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 highly composite numbers represent those numbers that set records in the divisor counting function A000005, and since the divisor is a special case of regular number, this sequence applies the "regular counting function" A010846 to terms in A002182.

Only 13 HCNs less than 36 * 10^6 are also "highly regular", i.e., appear in A244052. The largest HCN that is also highly regular is 27720, the 25th HCN and the 47th highly regular number.

Only 2 and 6 set records for the ratio A010846(n)/A000005(n).

Conjectures:

Let "tier" t consist of all terms A002110(t) <= m < A002110(t + 1) in A244052, where all such m in tier t have A001221(m) = t. The intersection of A002182 and A244052 is finite, consisting of 13 terms: {1, 2, 4, 6, 12, 24, 60, 120, 180, 840, 1260, 1680, 27720}. All of these terms are also in A060735 and not in A288813, as the latter are squarefree and have "gaps" among prime divisors. This intersection has the following number of terms in the "tiers" 0 through 5 of A244052: {1, 2, 3, 3, 3, 1}. If we look at A060735 as a number triangle T(n,k) = k * A002110(n) with 1 <= k < prime(n + 1), the terms are:

{0, 1},

{{1,1}, {1,2}},

{{2,1}, {2,2}, {2,4}},

{{3,2}, {3,4}, {3,6}},

{{4,4}, {4,6}, {4,8}},

{5,12}.

Conjecture: n = 30 is the largest number for which a(n) = -1.

For n > 2, a(n) is composite, since A010846(p) = 2 for prime p.

For n <= 3, a(n) = 2^n; for n > 3, a(n) < 2^n, and a(n) is in A024619.

Smallest k with omega(k) = i is A002110(i).

Conjecture: there are only 8 powerful terms (i.e., in A001694) in the sequence.

a(1) = A384000(3) = 1001; A010846(1001) = A024718(3) = 15; 1001 is the smallest number k with 3 distinct prime factors that has the smallest possible number of terms in row k of A162306, i.e., m <= k such that rad(m) | k.

For n > 30, 6 | a(n).

Let 1 <= r <= n be a number "regular to" n, that is, a product of prime divisors p that also divide n.

This sequence includes numbers n such that there are at least as many regulars r among the range 1 <= m <= n as m that are nonregular to n.

Divisors d are a special case of regular r such that d also divides n in addition to all prime divisors p of d also dividing n.

With the exception of r = 1, all regular numbers r are part of the cototient of n.

For prime n, A010846(n) = A000005(n). Composite n > 4 can have nondivisor r. The only prime numbers in a(n) are {2,3}.

Regular r divides some power of n, but reliably divides n^r. When r is a divisor d of n, r | n.

Because the divisors of n are a subset of the regulars of n, it is interesting to compare A010846(n) with A000005(n) = tau(n):

A010846(n) = n for n = (1,2). Tau(n) = n for n = (1,2).

n/2 < A010846(n) < n for n = (3,4,6,10,12,18,30). n/2 < tau(n) < n for n = (3,4,6).

A010846(n) = n/2 for n = 8. Tau(n) = n/2 for n = (8,12).

Sequence A020490 includes numbers n such that tau(n) >= phi(n). The number 24 is in A020490 but not in a(n). This is because A010846(24) = 11 (cf. A162306(24) = {1,2,3,4,6,8,9,12,16,18,24}).

Numbers n such that tau(n) >= n/2: {1,2,3,4,6,8,12}.