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If k is a term, then A348692(k) lists integers m such that k$ / m! is a square; and for each k, there exist only one (A349080) or two (A349081) such integers m.

See A348692 for further information, links and references about Olympiads.

Except for 1, all terms are even, and, when k is such an even term, corresponding m belong(s) to {k/2 - 2, k/2 - 1, k/2, k/2 + 1, k/2 + 2}.

This sequence is the union of {1} and of three infinite and disjoint subsequences:

-> A008586, so every positive multiple of 4 is a term and in this case, for k=4*q, (k$)/(k/2)! = ( 2^(k/4) * Product_{j=1..k/2} ((2j-1)!) )^2 (see example 4).

-> A060626, so every k = 4*q^2 - 2 (q >= 1) is a term (see examples 2 and 14).

-> 2*A055792 = {k = 2q^2 with q>1 in A001541} = {18, 578, ...} (see example 18).

This sequence is the union of three infinite and disjoint subsequences:

-> Numbers k = 8t^2 > 0 (A139098); for these numbers, m_1 = k/2 - 1 = 4t^2-1 < m_2 = k/2 = 4t^2 (see example for k = 8).

-> Numbers k = 8t*(t+1) (A035008); for these numbers, m_1 = k/2 = 4t(t+1) < m_2 = k/2 + 1 = (2t+1)^2 (see example for k = 16).

-> Even numbers of the form 2t^2-4, t>1 in A001541 (A349766); for these numbers, m_1 = k/2 + 1 = t^2 - 1 < m_2 = k/2 + 2 = t^2 (see example for k = 14).

See A348692 for further information.

Equivalently: numbers k for which there exists only one integer m with here m = k/2 + 1 such that A000178(k) / m! is a square, where A000178(k) = k$ = 1!*2!*...*k! is the superfactorial of k.

Equivalently: integers k such that k$ / (k/2+1)! and k$ / (k/2+2)! are both squares when A000178 (k) = k$ = 1!*2!*...*k! is the superfactorial of k (see A348692 for further information).

The 3 subsequences of A349081 are A035008, A139098 and this one.

This sequence can be calculated by a recursive algorithm:

Let B1 be an array of finite length, the "1" denotes that it is the first generation. Let B1' be the reversed version of B1. Let C be the element-wise product C = B1 * B1'. Then B2 is a concatenation of taking each element of B1 and add all divisors of the corresponding element in C. If we start with B1 = {1} then we get this sequence of arrays: B2 = {2}, B3 = {3, 4, 6}, ... . a(n) is the length of the array Bn. In short the length of Bn+1 and so a(n+1) is the sum over A000005(Bn * Bn').

The transform used in the definition of this sequence is its own inverse, so if c = S(b) then b = S(c). The eigensequence is 2^n = S(2^n).

There exist some transformation pairs of infinite sequences in the database:

A000027 <--> A000142; A000079 <--> A000079; A000045 <--> A001654;

A026549 <--> A038754; A100071 <--> A001405; A058295 <--> A------;

A111286 <--> A098011; A093968 <--> A205825; A166447 <--> A------;

A098558 <--> A004277; A010551 <--> A087811; A100538 <--> A329227;

A079352 <--> A------; A082458 <--> A------; A008233 <--> A264635;

A138278 <--> A------; A006501 <--> A264557; A336496 <--> A------;

A019464 <--> A------; A062112 <--> A------; A171647 <--> A359039;

A279312 <--> A------; A031923 <--> A------.

These transformation pairs are conjectured:

A137326 <--> A------; A066332 <--> A300902; A208147 <--> A308546;

A057895 <--> A------; A349080 <--> A------; A019442 <--> A------;

A349079 <--> A------.

("A------" means not yet in the database.)

Some sequences in the lists above may need offset adjustment to force a beginning with 1,2,... in the transformation.

If we allowed signed rational numbers, further interesting transformation pairs could be observed. For example, 1/n will transform into factorials with alternating sign. 2^(-n) transforms into ones with alternating sign and 1/A000045(n) into A000045 with alternating sign.