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A207331 Array of indices (or logarithms) Modd p for odd numbers smaller than p relative to basis of smallest primitive root. 0
0, 0, 0, 1, 0, 1, 2, 0, 1, 3, 4, 2, 0, 2, 3, 1, 4, 5, 0, 1, 5, 3, 2, 7, 4, 6, 0, 1, 4, 6, 2, 3, 8, 5, 7, 0, 1, 9, 6, 2, 4, 5, 10, 8, 3, 7, 0, 1, 10, 8, 2, 5, 12, 11, 7, 13, 9, 4, 6, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

For Modd n (not to be confused with mod n) see a comment on A203571.

The row lengths sequence for this array is 1 for row no. 1 and (p(n)-1)/2 with p(n):=A000040(n) (the primes).

For the definition of the index of a reduced number a mod n (but here we use Modd n) relative to a primitive root mod n, see, e.g., the Apostol reference, p. 213, and the tables on pp. 216-7. This mod n array is found under A054503 if the smallest primitive root mod n is taken as base. Because of its properties the index ind_b(a) is also called log_b(a), with the base b.

Here for Modd n, n>=2, primitive roots exist only for the values n with A206550(n)>0. There the smallest positive primitive roots, called here B(n) are also found. The allowed n values are shown in A206551. The indices Modd p(n), p(n):=A000040(n) (the primes) are called Ind_B(p(n))(a), with the odd numbers a smaller than p(n): 2*m-1=1,3,...,p(n)-2, for m=1,2,..., (p(n)-1)/2.

For odd p(n) the index Ind_B(p(n))(2*m-1) is defined as the unique value k from {0,1,...,(p(n)-3)/2}, such that B(p(n))^k = 2*m-1, with the base B(p(n)) the smallest positive primitive root Modd p(n).

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.

LINKS

Table of n, a(n) for n=1..60.

FORMULA

a(n,m) = Ind_B(p(n))(2*m-1), m=1,2,..., (p(n)-1)/2, n>=1. See the comment section for the definition of Ind_B(a).

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:   0

2,   3:   0

3,   5:   0   1

4,   7:   0   1   2

5,  11:   0   1   3  4  2

6,  13:   0   2   3  1  4   5

7,  17:   0   1   5  3  2   7   4   6

8,  19:   0   1   4  6  2   3   8   5    7

9,  23:   0   1   9  6  2   4   5  10    8    3   7

10, 29:   0   1  10  8  2   5  12  11    7   13   9   4   6  3

...

a(6,5) =4 because the base B(13) is here A206550(13)=7, and 7^4 = 2401, 2401 (Modd 13) := 2401 (mod 13) = 9 = 2*5-1.

CROSSREFS

Cf. A054503 (mod n case).

Sequence in context: A049600 A318602 A004542 * A134405 A091612 A253672

Adjacent sequences:  A207328 A207329 A207330 * A207332 A207333 A207334

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Mar 27 2012

STATUS

approved

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Last modified March 25 08:15 EDT 2019. Contains 321469 sequences. (Running on oeis4.)