A207330 Array of the orders Modd p, p a prime.

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

Author

Wolfdieter Lang, Mar 27 2012

Keywords

• nonn
• easy
• tabf

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).

Examples

			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).

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.