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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
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
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 (factorization 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
KEYWORD
nonn
AUTHOR
Cyril Banderier, Mar 05 2019
STATUS
approved