

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
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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 BanderierBarilMoreira article, they have many noteworthy arithmetical properties (proven in the BanderierLuca article).


LINKS

Table of n, a(n) for n=2..74.
Cyril Banderier, JeanLuc Baril, Céline Moreira Dos Santos, Right jumps in permutations, DMTCS 18:2#12, p. 117, 2017.
Cyril Banderier, Florian Luca, On the period mod m of polynomiallyrecursive sequences: a case study, arXiv:1903.01986 [math.NT], 2019.


FORMULA

The BanderierLuca 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 (p1).
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 "lcmmultiplicative" 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



