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Restricted growth sequence transform of A239968.

In the following, A stands for this sequence, A305800, and S -> T (where S and T are sequence A-numbers) indicates that for all i, j: S(i) = S(i) => T(i) = T(j).

For example, the following implications hold:

A -> A300247 -> A305897 -> A077462 -> A101296,

A -> A290110 -> A300250 -> A101296.

Call two numbers equivalent if they have the same prime factorization exponents (in the same order). This sequence enumerates the equivalence classes.

A055932(a(n)) = A071364(n). - David Wasserman, Dec 21 2004

From Antti Karttunen, Jun 13 2018: (Start)

After a(0) = 0, this is the restricted growth sequence transform of A071364. The latter sequence is an "ordered variant" of A046523, and because A101296 is the rgs-transform of A046523, it follows that for all i, j: a(i) = a(j) => A101296(i) = A101296(j).

(End)

This sequence gives a coarser partitioning of natural numbers than A290110, and finer than A101296:

For all i, j:

A290110(i) = A290110(j) => a(i) = a(j) => A101296(i) = A101296(j).

Also indices of the powers of 2 in A033676.

Also numbers in increasing order from the columns k of A163280 where k is a power of 2.

Observation: at least the first 82 terms of the subsequence of terms with no middle divisors (that is 3, 5, 7, 10, ...) coincide with at least the first 82 terms of A246955.

For the definition of middle divisor see A067742.

From Peter Munn, Oct 26 2023: (Start)

Most of the early terms are in A342081, which consists of powers of 2 together with products of a prime and a power of 2 where the prime is the larger. The exceptions are 24, 72, 80, 96, 112, ... .

The odd terms clearly consist of 1 and the odd primes. We can fully characterize the even terms by their A290110 values, which depend on the relative sizes of a number's divisors. A290110 provides a refinement of the classification of numbers by prime signature (cf. A212171): see the example below for numbers with the same prime signature as 48.

(End)

A number k does not have a strictly superior squarefree divisor if and only if k is at least as large as the square of rad(k), the largest squarefree divisor of k. All powerful numbers (A001694) have this property. This sequence lists the other such numbers.

Let rad(k) = A007947(k), the largest squarefree divisor, i.e., the squarefree kernel of k. A341645 lists the numbers without a strictly superior squarefree divisor.

A341645 = { k : rad(k) <= k/rad(k) } = { k : A007947(k) <= A003557(k) }, and it is evident that rad(k) <= k/rad(k) is true for powerful k, that is, k in A001694.

Since A001694 contains A001597, the above is also true for perfect powers k; A001597 is a proper subset of A341645.

This sequence contains "weak" k (in A052485) such that rad(k) < k/rad(k).

The presence of a number, k, in this sequence depends only upon A290110(k), i.e., upon the factorization pattern of its sequence of divisors as defined in A191743.

Let S = A006939 and let P = A002110. Almost all superprimorials are in this sequence: S \ {1, 2, 12, 360} is a proper subset. S(i) = S(i-1)*P(i), where S(i-1) = A003557(S(i)) and P(i) = rad(S(i)), and for i > 4, S(i-1) > P(i). Since prime(i) | S(i) but prime(i)^2 does not divide S(i), S(i) is not powerful. Corollary: almost all superprimorials are in A341645, since this sequence is a proper subset of A341645.

Numbers whose number of divisors of n (A000005) is equal to 3 + the number of prime factors of n (with multiplicity, A001222), and the third smallest divisor is a square of a prime (A001248).

This is a permutation of the positive integers.

From Peter Munn, May 18 2025: (Start)

Numbers with the same factorization pattern of their sequence of divisors (see A290110) and the same parity appear here in the same column.

For example, each column k > 2 includes the subsequence 2^(k-2) * p for all prime p > 2^(k-2).

(End)

Numbers whose number of divisors of n (A000005) is equal to 3 + the number of prime factors of n (with multiplicity, A001222), and the fourth smallest divisor is a square of a prime (A001248).

Also the number of regions in the 0 < x < y sector of the plane that are delimited by the lines with equations i*x + j*y = 0, where i and j are integers, not both 0, and |i| <= m, |j| <= n. This remark is motivated by Factorization Patterns (FPs) and Factorization Patterns of Sequences of Divisors (FPSD) concerns, as defined in A191743 and A290110. This is the case k=2 of a more general problem where k is omega(z)=A001221(z), the number of distinct primes dividing z, for which we would define T(n1,n2,...,nk) instead of T(m,n). The idea is the following: two numbers (e.g., 12 and 20) can have the same FP (p^2*q) without having the same FPSD ([1 < p < q < p^2 < p*q < p^2*q] != [1 < p < p^2 < q < p*q < p^2*q]). T(m,n) tells how many distinct FPSDs share the same FP of the p^m*q^n form. See the illustration for (m,n) = (2,1), section Links.