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A207330
Array of the orders Modd p, p a prime.
0
1, 1, 1, 2, 1, 3, 3, 1, 5, 5, 5, 5, 1, 3, 2, 6, 3, 6, 1, 8, 8, 8, 4, 8, 2, 4, 1, 9, 9, 3, 9, 3, 9, 9, 9, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 14, 7, 7, 7, 14, 7, 14, 2, 14, 14, 7, 7, 14, 1, 15, 3, 15, 15, 15, 15, 5, 15, 15, 15, 5, 3, 5, 5, 1, 9, 18, 9
OFFSET
1,4
COMMENTS
For Modd n (not to be confused with mod n) see a comment on A203571.
The row lengths sequence of this array is 1 for row n=1, and (p(n)-1)/2, with p(n):=A000040(n) (the primes), for row n>1.
A primitive root has order delta(p) = (p-1)/2 (delta is given by A055034).
FORMULA
a(n,m) = (multiplicative) order Modd p(n) of 2*m-1, for m=1,...,(p(n)-1)/2, with p(n):= A000040(n) (the primes), n>1, and for a(1,1) = 1 for the prime 2.
EXAMPLE
n, p(n)/m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
2m-1: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 ...
1, 2: 1
2, 3: 1
3, 5: 1 2
4, 7: 1 3 3
5, 11: 1 5 5 5 5
6, 13: 1 3 2 6 3 6
7, 17: 1 8 8 8 4 8 2 4
8, 19: 1 9 9 3 9 3 9 9 9
9, 23: 1 11 11 11 11 11 11 11 11 11 11
10, 29: 1 14 7 7 7 14 7 14 2 14 14 7 7 14
...
a(6,4) = 6 because 7^1 = 7, 7^2 = 49, 49 (Modd 13) := -49 (mod 13) = 3, 7^3 == 7*3 = 21,
21 (Modd 13) := -21 (mod 13) = 5, 7^4 == 7*5 = 35, 35 (Modd 13) = 35 (mod 13) = 9,
7^5 == 7*9=63, 63 (Modd 13):= 63 (mod 13) = 11, 7^6 == 7*11 = 77, 77 (Modd 13) := -77 (mod 13) = 1.
Row n=5: all 2*m-1, m>1, are primitive roots. The smallest positive one is 3.
Row n=6: only 7 and 11 are primitive roots. The smallest one is 7.
CROSSREFS
Cf. A086145 (mod n case).
Sequence in context: A274705 A257243 A097351 * A048600 A100578 A061315
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Mar 27 2012
STATUS
approved