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A207331
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Array of indices (or logarithms) Modd p for odd numbers smaller than p relative to basis of smallest primitive root.
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0
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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
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OFFSET
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1,7
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COMMENTS
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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).
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1986.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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.
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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