A323017 T(n,k) = A321477(n,k)/A321476(n,k), 0 <= k <= n - 1.

1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 4, 4, 4, 4, 1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 4, 2, 4, 2, 1, 2, 4, 2, 4, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2

Offset: 1

Author

Jianing Song, Jan 07 2019

Keywords

• nonn
• tabl

Comments

For Lucas sequences, say, rows in A172236, we are mainly concerned about the periods, ranks and the ratios of the periods to the ranks of them modulo a given integer n. The period of {A172236(k,m) modulo m} is given as A321477(n,k), and the rank, which is defined as the smallest l > 0 such that n divides A172236(k,l), is given as A321476(n,k). T(n,k) is their ratio, which is the multiplicative order of A172236(k,A321476(n,k)+1) modulo n.

T(n,k) has value 1, 2 or 4. This is because A172236(k,m+1)^4 == 1 (mod A172236(k,m)). For n > 2, T(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. See A172236 for some further properties.

Let p be an odd prime. If p == 3 (mod 4), the p^e-th row consists of only 1 and 2; if p == 5 (mod 8), the p^e-th row consists of only 1 and 4.

Examples

			Table begins
  1,
  1, 1,
  1, 2, 2,
  1, 1, 1, 1,
  1, 4, 4, 4, 4,
  1, 2, 2, 1, 2, 2,
  1, 2, 1, 2, 2, 1, 2,
  1, 2, 1, 2, 1, 2, 1, 2,
  1, 2, 2, 1, 2, 2, 1, 2, 2,
  1, 4, 2, 4, 2, 1, 2, 4, 2, 4,
  ...
		

Crossrefs

Cf. A172236, A321476, A321477.

Programs

PARI

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