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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

Table of n, a(n) for n=1..88.

FORMULA

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).

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

Cf. A000010, A038566, A174824.

Sequence in context: A319702 A162908 A222817 * A328368 A052297 A297162

Adjacent sequences:  A309425 A309426 A309427 * A309429 A309430 A309431

KEYWORD

nonn,tabf

AUTHOR

Jianing Song, Sep 18 2019

STATUS

approved

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Last modified November 17 23:26 EST 2019. Contains 329242 sequences. (Running on oeis4.)