

A321477


Regular triangle read by rows: T(n,k) is the period of {A172236(k,m)} modulo n, 0 <= k <= n  1.


2



1, 2, 3, 2, 8, 8, 2, 6, 4, 6, 2, 20, 12, 12, 20, 2, 24, 8, 6, 8, 24, 2, 16, 6, 16, 16, 6, 16, 2, 12, 8, 12, 4, 12, 8, 12, 2, 24, 24, 6, 8, 8, 6, 24, 24, 2, 60, 12, 12, 20, 6, 20, 12, 12, 60, 2, 10, 24, 8, 10, 24, 24, 10, 8, 24, 10, 2, 24, 8, 6, 8, 24, 4, 24, 8, 6, 8, 24
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OFFSET

1,2


COMMENTS

The period of {A172236(k,m)} modulo n is the smallest l such that A172236(k,m) == A172236(k,m+l) (mod n) for every m >= 0. Clearly, T(n,k) is divisible by A321476(n,k). Actually, the ratio is always 1, 2 or 4.
Though {A172236(0,m)} is not defined, it can be understood as the sequence 0, 1, 0, 1, ... So the first column of each row (apart from the first one) is always 2.
Every row excluding the first term is antisymmetric, that is, T(n,k) = T(n,nk) for 1 <= k <= n  1.
T(n,k) is the LCM of A321476(n,k) and the multiplicative order of (k + sqrt(k^2 + 4))/2 modulo n, where the multiplicative order of u modulo z is the smallest positive integer l such that (u^l  1)/z is an algebraic integer.


LINKS

Table of n, a(n) for n=1..78.


FORMULA

Let p be an odd prime. (i) If ((k^2+4)/p) = 1: if p == 1 (mod 4), then T(p^e,k) is divisible by p^(e1)*(p  1), and T(p^e,k) is even; if p == 3 (mod 4), then T(p^e,k) is divisible by p^(e1)*(p  1) but not divisible by p^(e1)*(p  1)/2. Here (a/p) is the Legendre symbol. (ii) If ((k^2+4)/p) = 1, then T(p^e,k) is divisible by 2*p^(e1)*(p + 1) but not divisible by p^(e1)*(p + 1). (iii) If k^2 + 4 is divisible by p, then T(p^e,k) = 4*p^e.
For e, k > 0, T(2^e,k) = 3*2^(e1) for odd k and 2^(ev(k,2)+1) for even k, where v(k,2) is the 2adic valuation of k.
If gcd(n_1,n_2) = 1, then T(n_1*n_2,k) = lcm(T(n_1,k mod n_1),T(n_2, k mod n_2)).
For n > 2, a(n,k)/A321476(n,k) = 4 iff A321476(n,k) is odd; 1 iff A321476(n,k) is even but not divisible by 4; 2 iff A321476(n,k) is divisible by 4.
Let p be an odd prime. (i) If ((k^2+4)/p) = 1: if p == 5 (mod 8), then T(p^e,k)/A321476(p^e,k) != 2; if p == 3 (mod 4), then T(p^e,k)/A321476(p^e,k) = 1. (ii) If ((k^2+4)/p) = 1: if p == 1 (mod 4), then T(p^e,k)/A321476(p^e,k) = 4; if p == 3 (mod 4), then T(p^e,k)/A321476(p^e,k) = 2.
T(n,k) <= 6*n.


EXAMPLE

Table begins
1;
2, 3;
2, 8, 8;
2, 6, 4, 6;
2, 20, 12, 12, 20;
2, 24, 8, 6, 8, 24;
2, 16, 6, 16, 16, 6, 16;
2, 12, 8, 12, 4, 12, 8, 12;
2, 24, 24, 6, 8, 8, 6, 24, 24;
2, 60, 12, 12, 20, 6, 20, 12, 12, 60;
...


PROG

(PARI) A172236(k, m) = ([k, 1; 1, 0]^m)[2, 1]
T(n, k) = my(i=1); while(A172236(k, i)%n!=0(A172236(k, i+1)1)%n!=0, i++); i


CROSSREFS

Cf. A172236, A321476 (ranks).
Sequence in context: A098513 A134347 A057761 * A209998 A163204 A299619
Adjacent sequences: A321474 A321475 A321476 * A321478 A321479 A321480


KEYWORD

nonn,tabl


AUTHOR

Jianing Song, Nov 11 2018


STATUS

approved



