STATUS
proposed
approved
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proposed
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editing
proposed
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, 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, 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, 8, 16, 16, 16, 4, 16, 8, 2,
18, 6, 18, 6, 18, 2,
19, 18, 18, 9, 9, 9, 3, 6, 9, 18, 3, 6, 18, 18, 18, 9, 9, 2,
20, 4, 4, 4, 10, 4, 4, 2,
...
...
proposed
editing
editing
proposed
For gcd(n,r) = 1, 1 <= r <= n, let d(n,r) is be the multiplicative order of {{r, 1}, {0, 1}}, then T(n,k) = d(n,A038566(k)).
approved
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proposed
approved
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proposed
For 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.
For gcd(n,r) = 1, 1 <= r <= n, let d(n,r) is the multiplicative order of {{r, 1}, {0, 1}}, then T(n,k) = d(n,A038566(k)).
(a) If p is an odd prime, then d(p^e,r) = p^e if r == 1 (mod p), ord(r,p^e) otherwise;
(b) d(2^e,r) = 2^(e+1-v2(r+1)), where v2(t) is the 2-adic valuation of t;
(c) For gcd(m,n) = 1, d(m*n,r) = lcm(d(m,r mod m),d(n,r mod n)).
The LCM of the n-th row is A174824(n).
For n = 14 and k = 4, let M = {{A038566(n,k), 1}, {0, 1}} = {{9, 1}, {0, 1}}, then:
- M^2 mod 14 = {{11, 10}, {0, 1}};
- M^3 mod 14 = {{1, 7}, {0, 1}};
- M^4 mod 14 = {{9, 8}, {0, 1}};
- M^5 mod 14 = {{11, 3}, {0, 1}};
- M^6 mod 14 = {{1, 0}, {0, 1}}.
So T(14,4) = d(14,9) = 6.
(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
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.
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, 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, 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
1,2
Table starts
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, 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, 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, 8, 16, 16, 16, 4, 16, 8, 2,
18, 6, 18, 6, 18, 2,
19, 18, 18, 9, 9, 9, 3, 6, 9, 18, 3, 6, 18, 18, 18, 9, 9, 2,
20, 4, 4, 4, 10, 4, 4, 2,
...
allocated
nonn,tabf
Jianing Song, Sep 18 2019
approved
editing