A306699 Periods of A265165(k) mod n.

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

Author

Cyril Banderier, Mar 05 2019

Keywords

• nonn

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).

Examples

			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.
		

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.

Crossrefs

Cf. A265163, A265164, A265165.

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)).