A309428 - OEIS

%I #52 Feb 10 2024 18:53:21

%S 1,2,3,2,4,2,5,4,4,2,6,2,7,3,6,3,6,2,8,4,8,2,9,6,9,6,9,2,10,4,4,2,11,

%T 10,5,5,5,10,10,10,5,2,12,4,6,2,13,12,3,6,4,12,12,4,3,6,12,2,14,6,6,6,

%U 6,2,15,4,6,12,4,10,12,2,16,8,16,4,16,8,16,2,17,8,16,4,16,16,16,8

%N Irregular triangle read by rows: T(n,k) is the multiplicative order of {{A038566(n,k), 1}, {0, 1}} modulo n, n >= 1, 1 <= k <= A000010(n).

%C Let M = {{r, 1}, {0, 1}}, then M^e = {{r^e, 1 + r + r^2 + ... + r^(e-1)}, {0, 1}}. As a result, for gcd(r, n) = 1, the multiplicative order of {{r, 1}, {0, 1}} modulo n is n if r == 1 (mod n) and ord(r,n*(r-1)) otherwise, where ord(r,t) is the multiplicative order of r modulo t.

%F For gcd(n,r) = 1, 1 <= r <= n, let d(n,r) be the multiplicative order of {{r, 1}, {0, 1}}, then T(n,k) = d(n,A038566(k)).

%F (a) If p is an odd prime, then d(p^e,r) = p^e if r == 1 (mod p), ord(r,p^e) otherwise;

%F (b) d(2^e,r) = 2^(e+1-v2(r+1)), where v2(t) is the 2-adic valuation of t;

%F (c) For gcd(m,n) = 1, d(m*n,r) = lcm(d(m,r mod m),d(n,r mod n)).

%F The LCM of the n-th row is A174824(n).

%e Table starts

%e   1,

%e   2,

%e   3, 2,

%e   4, 2,

%e   5, 4, 4, 2,

%e   6, 2,

%e   7, 3, 6, 3, 6, 2,

%e   8, 4, 8, 2,

%e   9, 6, 9, 6, 9, 2,

%e   10, 4, 4, 2,

%e   11, 10, 5, 5, 5, 10, 10, 10, 5, 2,

%e   12, 4, 6, 2,

%e   13, 12, 3, 6, 4, 12, 12, 4, 3, 6, 12, 2,

%e   14, 6, 6, 6, 6, 2,

%e   15, 4, 6, 12, 4, 10, 12, 2,

%e   16, 8, 16, 4, 16, 8, 16, 2,

%e   17, 8, 16, 4, 16, 16, 16, 8, 8, 16, 16, 16, 4, 16, 8, 2,

%e   18, 6, 18, 6, 18, 2,

%e   19, 18, 18, 9, 9, 9, 3, 6, 9, 18, 3, 6, 18, 18, 18, 9, 9, 2,

%e   20, 4, 4, 4, 10, 4, 4, 2,

%e   ...

%e For n = 14 and k = 4, let M = {{A038566(n,k), 1}, {0, 1}} = {{9, 1}, {0, 1}}, then:

%e - M^2 mod 14 = {{11, 10}, {0, 1}};

%e - M^3 mod 14 = {{1, 7}, {0, 1}};

%e - M^4 mod 14 = {{9, 8}, {0, 1}};

%e - M^5 mod 14 = {{11, 3}, {0, 1}};

%e - M^6 mod 14 = {{1, 0}, {0, 1}}.

%e So T(14,4) = d(14,9) = 6.

%o (PARI) row(n) = my(v=vector(n,i,i),u=vector(eulerphi(n),i,n)); v=select(i->gcd(n,i)==1,v); for(i=2, #v, u[i]=znorder(Mod(v[i], n*(v[i]-1)))); u

%Y Cf. A000010, A038566, A174824.

%K nonn,tabf

%O 1,2

%A _Jianing Song_, Sep 18 2019