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A309428
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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).
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0
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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