cp's OEIS Frontend

The set of b values (see formula), and therefore also a(n), depends only on the prime signature of n. So, for example, a(24) will be identical to a(n) of any other n which is also of the form p_1^3*p_2, (e.g., 40, 54, 56).

The value of b_1 will always be 1. When n is prime, the only nonzero b will be b_1, so therefore a(n) will be 1.

In any arrangement, both dice will have a 0, and one will have a 1 (here called the lead die). To determine any one of the actual arrangements to numbers on the dice, select one of the permutations of divisors (for the lead die), then select another permutation that is either the same length as that of the lead die, or one less. For example, if n = 12, we might select 2*3*2 for the lead die, and 3*4 for the second die. These numbers effectively tell you when to "switch track" when numbering the dice, and will uniquely result in the numbering: (0,1,6,7,12,13,72,73,78,79,84,85; 0,2,4,18,20,22,36,38,40,54,56,58).

a(n) is the number of (unordered) pairs of polynomials c(x) = x^c_1 + x^c_2 + ... + x^c_n, d(x) = x^d_1 + x^d_2 + ... + x^d_n with nonnegative integer exponents such that c(x)*d(x) = (x^(n^2)-1)/(x-1). - Alois P. Heinz, May 13 2016

a(n) is also the number of principal reversible squares of order n. - S. Harry White, May 19 2018

From Gus Wiseman, Oct 29 2021: (Start)

Also the number of ordered factorizations of n^2 with alternating product 1. This follows from the author's formula. Taking n instead of n^2 gives a(sqrt(n)) if n is a perfect square, otherwise 0. Here, an ordered factorization of n is a sequence of positive integers > 1 with product n, and the alternating product of a sequence (y_1,...,y_k) is Product_i y_i^((-1)^(i-1)). For example, the a(1) = 1 through a(9) = 3 factorizations are:

() (22) (33) (44) (55) (66) (77) (88) (99)

(242) (263) (284) (393)

(2222) (362) (482) (3333)

(2233) (2244)

(2332) (2442)

(3223) (4224)

(3322) (4422)

(22242)

(24222)

(222222)

The even-length case is A347464.

(End)

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

First differs from A348383 at a(216) = 27, A348383(216) = 28.

A factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

These permutations are ordered factorizations of n with no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z.

The version without twins for n > 0 is a(n) + 1 if n is a perfect square; otherwise a(n).

Length of row n equals A001221(n).

The multiset of positive exponents in n's prime factorization completely determines a(n) for a host of OEIS sequences, including several "core" sequences. Of those not cross-referenced here or in A212172, many can be found by searching the database for A025487.

(Note: Differing opinions may exist about whether the prime signature of n should be defined as this multiset itself, or as a symbol or collection of symbols that identify or "signify" this multiset. The definition of this sequence is designed to be compatible with either view, as are the original comments. When n >= 2, the customary ways to signify the multiset of exponents in n's prime factorization are to list the constituent exponents in either nonincreasing or nondecreasing order; this table gives the nonincreasing version.)

Table lists exponents in the order in which they appear in the prime factorization of a member of A025487. This ordering is common in database comments (e.g., A008966).

Each possible multiset of an integer's positive prime factorization exponents corresponds to a unique partition that contains the same elements (cf. A000041). This includes the multiset of 1's positive exponents, { } (the empty multiset), which corresponds to the partition of 0.

Differs from A124010 from a(23) on, corresponding to the factorization of 18 = 2^1*3^2 which is here listed as row 18 = [2, 1], but as [1, 2] (in the order of the prime factors) in A124010 and also in A118914 which lists the prime signatures in nondecreasing order (so that row 12 = 2^2*3^1 is also [1, 2]). - M. F. Hasler, Apr 08 2022

An ordered factorization of n is a finite sequence of positive integers > 1 with product n.

We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.

From Gus Wiseman, Mar 05 2021: (Start)

This sequence counts all of the following essentially equivalent things:

1. Chains of distinct inferior divisors from n to 1, where a divisor d|n is inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.

2. Chains of divisors from n to 1 whose first-quotients (in analogy with first-differences) are term-wise greater than or equal to their decapitation (maximum element removed). For example, the divisor chain q = 60/4/2/1 has first-quotients (15,2,2), which are >= (4,2,1), so q is counted under a(60).

3. Chains of divisors from n to 1 such that x >= y^2 for all adjacent x, y.

4. Factorizations of n where each factor is greater than or equal to the product of all previous factors.

(End)

An alternative wording: Number of chains of divisors starting with n and having all adjacent parts x > y^2.