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 A321478 Regular triangle read by rows: T(n,k) is the rank of {A316269(k,m)} modulo n, 0 <= k <= n - 1. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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. LINKS 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. EXAMPLE 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;   ... PROG (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 CROSSREFS Cf. A316269, A321479 (periods). Sequence in context: A282162 A276857 A244893 * A076982 A283617 A164886 Adjacent sequences:  A321475 A321476 A321477 * A321479 A321480 A321481 KEYWORD nonn,tabl AUTHOR Jianing Song, Nov 11 2018 STATUS approved

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Last modified June 24 05:39 EDT 2019. Contains 324318 sequences. (Running on oeis4.)