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Restricted growth sequence transform of A291752.

For all i, j:

a(i) = a(j) => A291751(i) = A291751(j),

a(i) = a(j) => A326199(i) = A326199(j) => A294877(i) = A294877(j),

a(i) = a(j) => A322021(i) = A322021(j),

a(i) = a(j) => A295888(i) = A295888(j),

a(i) = a(j) => A296090(i) = A296090(j).

Row 2 records the primes (A000040). Rows 3 and 4 record the semiprimes (A001358). Rows 5, 6 and 9 record the 3-almost primes (A014612) etc. A058933 is a similar sequence based on k-almost primes.

The graph of this sequence is interesting for large n because it shows multiple curves, one for each prime signature. For example, the six highest curves on the graph of a(n) for n up to 10^4 are for the (1,1), (1,1,1), (1), (2,1,1), (2,1), and (1,1,1,1) prime signatures. The (1) curve dominates until n=58; the (1,1) curve dominates until n=1279786, when the (1,1,1) curve intersects the (1,1) curve. Each (1,1,...,1) curve dominates for a finite number of n.

Ordinal transform of A101296. - Antti Karttunen, May 15 2017

a(n) is the number of positive integers up to n with the same prime signature as n. For example, the a(20) = 3 numbers are {12, 18, 20}. - Gus Wiseman, Jul 08 2019

Ordinal transform of A046523. - Alois P. Heinz, May 31 2020

First differs from A296120 at a(36) = 15, A296120(36) = 14. - Gus Wiseman, Apr 27 2025

Each "part" in parentheses is distinct from all others at the same level. Thus (3*2)*(2) is allowed but (3)*(2*2) and (3*2*2) are not.

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

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)

A101296(a(n)) gives a permutation of natural numbers.

As A065043 is not multiplicative, neither is this sequence.

For all numbers n with an odd number of prime factors (with mult.), a(n) = 0. Proof: In the convolution formula, when n is any term of A026424, either the divisor (n/d) or d (but not both) has an odd number of prime factors. As A065043 is zero for all A026424, it is easy to show by induction that also a(n) is zero for all such numbers. Therefore, nonzero values (including any odd values, see A359765) occur only on a subset of A028260, and A359764(n) <= A065043(n).

The permutation satisfies A008578(a(n)) = a(A008578(n)) for all n, and is self-inverse.

The sequence of fixed points begins as 0, 1, 6, 11, 29, 36, 66, 95, 107, 121, 149, 174, 216, 313, 319, 396, 427, ... and is itself multiplicative in a sense that if a and b are fixed points, then also a*b is a fixed point.

The records are 0, 1, 3, 9, 19, 61, 281, 361, 843, 1159, 1811, 3721, 5339, 5433, 17141, 78961, 110471, 236883, 325679, ...

and they occur at positions 0, 1, 2, 4, 5, 7, 13, 25, 26, 35, 37, 49, 65, 74, 91, 169, 259, 338, 455, ...

(Note how the permutations map squares to squares, and in general keep the prime signature the same.)

Composition with similarly constructed A235199 gives the permutations A234743 & A234744 with more open cycle-structure.

The result of applying a permutation of the prime numbers to the prime factors of n. - Peter Munn, Dec 15 2019

Per the formula, a(n) = 1 at prime n. As the sequence extends, additional values become more frequent than 1. These values can be characterized, for example, a(n) = 19 is seen at n corresponding to A007304, a(n) = 3 is seen at n corresponding to A006881, a(n) = 113 is seen at n corresponding to A085987. - Bill McEachen, Dec 28 2023

From Antti Karttunen, Jan 02 2024: (Start)

The value of a(n) depends only on the prime signature of n. In other words, for all i, j >= 1, it holds that A101296(i) = A101296(j) => a(i) = a(j). Moreover, it seems that the converse proposition also holds, that for all i, j >= 1, a(i) = a(j) => A101296(i) = A101296(j), i.e., for each distinct prime signature there exists a distinct value of a(n). This has been empirically checked up to the first 21001 prime signatures in A025487 (see A366884), and can be proved if one can show that the latter sequence (equally: A366377) is injective. If this conjecture holds, it would imply an unlimited number of statements like those given in the previous comment (see the formula section of A101296).

Questions: Are there any terms of the form 10k+4 or 10k+6? What is the asymptotic density of terms of the form 10k+5 (those ending with digit "5")? Compare to the data shown in A366884.

For squarefree n > 1, a(n) is never even, and apparently, never a multiple of five. See comments in A052886.

(End)

The definition for the factorization pattern of the sequence of divisors of a number n is the same as in sequence A191743. Let's use the abbreviation FPSD. One can generate a list of distinct FPSD by trying all integers, 1, 2, 3, ..., and ignoring duplicates. a(n) is the index of the FPSD of n in this list.

From Antti Karttunen, Mar 07 & 08 2018: (Start)

This is NOT restricted growth sequence transform of A297174, but instead A300250 is, from which this differs for the first time at n=858, where a(858) = 115, while A300250(858) = 75.

This gives a finer partitioning of natural numbers than A300250, and indeed we have:

For all i, j:

a(i) = a(j) => A300250(i) = A300250(j) => A101296(i) = A101296(j).

(End)