A321478 Regular triangle read by rows: T(n,k) is the rank of {A316269(k,m)} modulo n, 0 <= k <= n - 1.

1, 2, 3, 2, 3, 3, 2, 3, 4, 3, 2, 3, 5, 5, 3, 2, 3, 6, 6, 6, 3, 2, 3, 7, 4, 4, 7, 3, 2, 3, 8, 3, 4, 3, 8, 3, 2, 3, 9, 6, 9, 9, 6, 9, 3, 2, 3, 10, 15, 6, 6, 6, 15, 10, 3, 2, 3, 11, 5, 5, 6, 6, 5, 5, 11, 3, 2, 3, 12, 6, 6, 3, 4, 3, 6, 6, 12, 3

Offset: 1

Author

Jianing Song, Nov 11 2018

Keywords

• nonn
• tabl

Comments

The rank of {A316269(k,m)} modulo n is the smallest l such that n divides A316269(k,l).

Though {A316269(0,m)} is not defined, it can be understood as the sequence 0, 1, 0, -1, 0, 1, 0, -1, ... So the first column of each row (apart from the first one) is always 2.

Though {A316269(1,m)} is not defined, it can be understood as the sequence 0, 1, 1, 0, -1, -1, 0, 1, 1, 0, -1, -1, ... So the second column of each row is always 3.

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 multiplicative order of ((k + sqrt(k^2 - 4))/2)^2 modulo n*sqrt(k^2 - 4), 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,  3,  3;
  2,  3,  4,  3;
  2,  3,  5,  5,  3;
  2,  3,  6,  6,  6,  3;
  2,  3,  7,  4,  4,  7,  3;
  2,  3,  8,  3,  4,  3,  8,  3;
  2,  3,  9,  6,  9,  9,  6,  9,  3;
  2,  3, 10, 15,  6,  6,  6, 15, 10,  3;
  ...
		

Crossrefs

Cf. A316269, A321479 (periods).

Programs

PARI

A316269(k, m) = ([k, -1; 1, 0]^m)[2, 1]
T(n, k) = my(i=1); while(A316269(k, i)%n!=0, i++); i

Formula

Let p be a prime >= 5. (i) If k^2 - 4 is not divisible by p, then T(p^e,k) is divisible by p^(e-1)*(p - ((k^2-4)/p))/2. Here (a/p) is the Legendre symbol. (ii) If k^2 - 4 is divisible by p, then T(p^e,k) = p^e.

For e >= 2 and 1 < k < 2^e - 1, T(2^e,k) = 3*2^(e-v(k^2-1,2)) for odd k and 2^(e-v(k,2)+1) for even k, where v(k,2) is the 2-adic valuation of k.

For e > 0 and k > 1, T(3^e,k) = 2*3^(e-v(k,3)) for k divisible by 3 and 3^(e-v(k^2-1,3)+1) otherwise.

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)).

T(n,k) <= (3/2)*n.