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A306699 Periods of A265165(k) mod n.

%I A306699 #27 Dec 11 2019 08:55:54
%S A306699 2,12,8,1,12,84,8,36,2,1,24,104,84,12,16,544,36,1,8,84,2,1012,24,1,
%T A306699 104,108,168,1,12,1,32,12,544,84,72,2664,2,312,8,1,84,3612,8,36,1012,
%U A306699 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
%N A306699 Periods of A265165(k) mod n.
%C A306699 Let b(k) be the sequence A265165(k).
%C A306699 a(n) = period({b(k) mod n}) = smallest p > 0 such that b(k+p) = b(k) mod n (for all large enough k).
%C A306699 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).
%H A306699 Cyril Banderier, Jean-Luc Baril, Céline Moreira Dos Santos, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/rightjumps.pdf">Right jumps in permutations</a>, DMTCS 18:2#12, p. 1-17, 2017.
%H A306699 Cyril Banderier, Florian Luca, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/BanderierLuca.pdf">On the period mod m of polynomially-recursive sequences: a case study</a>, arXiv:1903.01986 [math.NT], 2019.
%F A306699 The Banderier-Luca article proves the following properties:
%F A306699 a(n) = 1 iff n is a product of primes in 0,1,4 mod 5.
%F A306699 a(n) = 2 iff n/2 is a product of primes in 0,1,4 mod 5.
%F A306699 If a(n) is not 1, then it is an even number.
%F A306699 For any prime p, a(p) | 2 p (p-1).
%F A306699 For any prime p not in 0,1,4 mod 5, (and p^r <> 4), a(p^r) = p^r a(p).
%F A306699 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)).
%e A306699 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.
%Y A306699 Cf. A265163, A265164, A265165.
%K A306699 nonn
%O A306699 2,1
%A A306699 _Cyril Banderier_, Mar 05 2019