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A309428 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). 0
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
LINKS
FORMULA
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)).
(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).
EXAMPLE
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,
...
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.
PROG
(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
CROSSREFS
Sequence in context: A162908 A222817 A344324 * A328368 A052297 A365049
KEYWORD
nonn,tabf
AUTHOR
Jianing Song, Sep 18 2019
STATUS
approved

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Last modified March 29 09:14 EDT 2024. Contains 371268 sequences. (Running on oeis4.)