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A321476 Regular triangle read by rows: T(n,k) is the rank of {A172236(k,m)} modulo n, 0 <= k <= n - 1. 2
1, 2, 3, 2, 4, 4, 2, 6, 4, 6, 2, 5, 3, 3, 5, 2, 12, 4, 6, 4, 12, 2, 8, 6, 8, 8, 6, 8, 2, 6, 8, 6, 4, 6, 8, 6, 2, 12, 12, 6, 4, 4, 6, 12, 12, 2, 15, 6, 3, 10, 6, 10, 3, 6, 15, 2, 10, 12, 4, 10, 12, 12, 10, 4, 12, 10, 2, 12, 4, 6, 4, 12, 4, 12, 4, 6, 4, 12 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The rank of {A172236(k,m)} modulo n is the smallest l such that n divides A172236(k,l).

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 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.

LINKS

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

FORMULA

Let p be an odd prime. (i) If k^2 + 4 is not divisible by p: if p == 1 (mod 4), then T(p^e,k) is divisible by p^(e-1)*(p - ((k^2+4)/p))/2; if p == 3 (mod 4), then T(p^e,k) is divisible by p^(e-1)*(p - ((k^2+4)/p)) but not 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 >= 3 and k > 0, T(2^e,k) = 3*2^(e-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.

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) <= 2*n.

EXAMPLE

Table begins

  1;

  2,  3;

  2,  4,  4;

  2,  6,  4,  6;

  2,  5,  3,  3,  5;

  2, 12,  4,  6,  4, 12;

  2,  8,  6,  8,  8,  6,  8;

  2,  6,  8,  6,  4,  6,  8,  6;

  2, 12, 12,  6,  4,  4,  6, 12, 12;

  2, 15,  6,  3, 10,  6, 10,  3,  6, 15;

  ...

PROG

(PARI) A172236(k, m) = ([k, 1; 1, 0]^m)[2, 1]

T(n, k) = my(i=1); while(A172236(k, i)%n!=0, i++); i

CROSSREFS

Cf. A172236, A321477 (periods).

Sequence in context: A215182 A214906 A274228 * A214567 A259363 A087437

Adjacent sequences:  A321473 A321474 A321475 * A321477 A321478 A321479

KEYWORD

nonn,tabl

AUTHOR

Jianing Song, Nov 11 2018

STATUS

approved

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Last modified June 21 12:48 EDT 2021. Contains 345364 sequences. (Running on oeis4.)