A321477 Regular triangle read by rows: T(n,k) is the period of {A172236(k,m)} modulo n, 0 <= k <= n - 1.

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

Offset: 1

Author

Jianing Song, Nov 11 2018

Keywords

• nonn
• tabl

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,n-k) 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.

Examples

			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;
  ...
		

Crossrefs

Cf. A172236, A321476 (ranks).

Programs

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

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^(e-1)*(p - 1), and T(p^e,k) is even; if p == 3 (mod 4), then T(p^e,k) is divisible by p^(e-1)*(p - 1) but not divisible by p^(e-1)*(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^(e-1)*(p + 1) but not divisible by p^(e-1)*(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^(e-1) for odd k and 2^(e-v(k,2)+1) for even k, where v(k,2) is the 2-adic 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.