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A306699 Periods of A265165(k) mod n. 0
2, 12, 8, 1, 12, 84, 8, 36, 2, 1, 24, 104, 84, 12, 16, 544, 36, 1, 8, 84, 2, 1012, 24, 1, 104, 108, 168, 1, 12, 1, 32, 12, 544, 84, 72, 2664, 2, 312, 8, 1, 84, 3612, 8, 36, 1012, 4324, 48, 588, 2, 1632, 104, 5512, 108, 1, 168, 12, 2, 1, 24, 1, 2, 252, 64, 104, 12, 2948, 544, 3036, 84, 1, 72, 10512, 2664 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

Let b(k) be the sequence A265165(k).

a(n) = period({b(k) mod n}) = smallest p > 0 such that b(k+p) = b(k) mod n (for all large enough k).

The sequences b(k) and a(n) were introduced in the Banderier-Baril-Moreira article, they have many noteworthy arithmetical properties (proven in the Banderier-Luca article).

LINKS

Table of n, a(n) for n=2..74.

Cyril Banderier, Jean-Luc Baril, Céline Moreira Dos Santos, Right jumps in permutations, DMTCS 18:2#12, p. 1-17, 2017.

Cyril Banderier, Florian Luca, On the period mod m of polynomially-recursive sequences: a case study, arXiv:1903.01986 [math.NT], 2019.

FORMULA

The Banderier-Luca article proves the following properties:

a(n) = 1 iff n is a product of primes in 0,1,4 mod 5.

a(n) = 2 iff n/2 is a product of primes in 0,1,4 mod 5.

If a(n) is not 1, then it is an even number.

For any prime p, a(p) | 2 p (p-1).

For any prime p not in 0,1,4 mod 5, (and p^r <> 4), a(p^r) = p^r a(p).

a(n) is an "lcm-multiplicative" sequence: a(n1*n2) = lcm(a(n1), a(n2)) (for n1,n2 coprime), this implies that if n = p1^e1 ... pk^ek (factorisation in distinct primes) then a(n) = lcm(a(p1^e1), ..., a(pk^ek)).

EXAMPLE

A265165(k) mod 15 = (10,5,10,10,0,10,5,10,5,5,0,5)... and this pattern of length 12 repeats, therefore a(15) = 12.

CROSSREFS

Cf. A265163, A265164, A265165.

Sequence in context: A133437 A245692 A182126 * A266511 A014964 A173181

Adjacent sequences:  A306696 A306697 A306698 * A306700 A306701 A306702

KEYWORD

nonn

AUTHOR

Cyril Banderier, Mar 05 2019

STATUS

approved

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Last modified May 24 14:49 EDT 2019. Contains 323532 sequences. (Running on oeis4.)