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Analogous to A002183.

Records transform of A010846. -Michael De Vlieger, Mar 08 2017

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

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.

Composite numbers m have nondivisors k in the cototient such that k | n^e with e > 1. These k appear in row n of A272618 and are enumerated by A243822(n). These nondivisors k are a kind of "regular" number along with divisors d of n; both are listed in row n of A162306 and are together enumerated by A010846(n).

This sequence lists numbers that have more nondivisors k in the cototient of n than divisors d.

This sequence contains all n for which A299990(n) is positive.

The smallest odd term is 165.

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

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

a(n) = terms t of row m of A288784 such that A002110(m) < t < 2*A002110(m).

The only odd term is 3; the only other term not ending in 10, 30, 70, or 90 in decimal is 42.

All terms t in row m have A001221(t) = m and at least one prime q coprime to t such that q < A006530(t).

Consider "tier" m and primorial p_m# = A002110(m), let "distension" i = pi(A006530(T(m, k))) - m and let "depth" j = m - pi(A053669(T(m, k))) + 1. Distension is the difference in the index of gpf(T(m, k)) and pi(m), while depth is the difference between the index of the least prime totative of T(m, k) and pi(m) + 1. We can calculate the maximum distension i given m and j via i_max = A020900(m - j + 1) - m - j + 1. This enables us to use permutations of 0 and 1 values in the notation A054841 and produce a(n) with some efficiency.

The most efficient method of generating a(n) is via f(x) = A287352(x), i.e., subtracting 1 from all values in row x of A287352. We use a pointer variable to direct increment on f(p_m#) = a constant array of m 1's, until we have exhausted producing terms p_m# < t < 2*p_m#. This enables the generation of T(m, k) for 1 <= m <= 100.

A010846(n) = A000005(n) + A243822(n).

Successive terms in this sequence represent increasing differences A243822(n) - A000005(n).

A000079 = records in -1 * A299990, since A243822(p^e)=0 for e>=0, n = 2^k sets records in A000005(n). The corresponding records are in A000027.

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.

From Michael De Vlieger, Nov 17 2017: (Start)

Terms in a(n) appear in A244052 except {78, 126, 990, 19110, 6276270, ...}.

Primorials A002110(t) seem to divide this sequence into "tiers" thus: all terms A002110(t) <= m < A002110(t + 1), wherein A001221(m) = t as seen in A244052.

Terms in A244052 appear in a(n) except {2, 4, 12, 24, 120, 180, 1260, 1680, 18480, 27720, 360360, ...}. These numbers seem to have significantly more divisors than terms that are slightly greater or lesser in a(n).

Conjecture: all terms of a(n) with n > 92 also appear in A244052, and all terms in A244052 greater than a(92) = 6276270 appear in a(n).

(End)

A060735 and A002110 are subsets.

This sequence is a necessary but insufficient condition for A244052. Terms that are in A060735 and A002110 are also in A244052. The first terms of this sequence that are not in A244052 are {3, 4290, 881790, 903210, 1009470, 17160990, 363993630, 380570190, 406816410, 434444010, ...}.

Primorial p_n# = A002110(n) is the smallest squarefree number with n prime factors. Consider the list of squarefree numbers t with n prime factors greater than and including A002110(n) but less than 2*A002110(n). Extend the list to include products k*t of this list with 1 <= k < prime(n+1) such that k*t < (k+1)*p_n#. This list contains squarefree numbers k*t with n distinct primes and presumes that the number (k+1)*p_n# serves as a "limit" beyond which k*t > (k+1)p_n# are not in the sequence.

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}.