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A212486 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(7) listed in ascending order. 6
1, 2, 3, 6, 4, 8, 12, 16, 24, 48, 9, 18, 19, 38, 57, 114, 171, 342, 5, 10, 15, 20, 25, 30, 32, 40, 50, 60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800, 1200, 2400, 2801, 5602, 8403, 16806, 36, 43, 72, 76, 86, 129, 144, 152, 172, 228, 258 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The elements m of row n, are also solutions to the equation: multiplicative order of 7 mod m = n, with gcd(m,7) = 1, cf. A053450.

REFERENCES

R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 560-562.

V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28

LINKS

Boris Putievskiy, Table of n, a(n) for n = 1..102

Eric Weisstein's World of Mathematics, Irreducible Polynomial

XIAO, (computes the order of an irreducible polynomial over a finite field GF(p))

FORMULA

T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (7^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,3,6}.

|M(n)| = Sum_{d|n} mu(n/d)*tau(7^d-1) = A059889(n).

EXAMPLE

Triangle T(n,k) begins:

1,  2,  3,  6;

4,  8, 12, 16, 24,  48;

9, 18, 19, 38, 57, 114, 171, 342;

5, 10, 15, 20, 25,  30,  32,  40, 50, 60, 75, 80, 96, 100, 120, 150, 160, 200, 240, 300, 400, 480, 600, 800, 1200, 2400;

MAPLE

with(numtheory):

M:= proc(n) option remember;

      `if`(n=1, {1, 2, 3, 6}, divisors(7^n-1) minus U(n-1))

    end:

U:= proc(n) option remember;

      `if`(n=0, {}, M(n) union U(n-1))

    end:

T:= n-> sort([M(n)[]])[]:

seq(T(n), n=1..7);

CROSSREFS

Cf. A053446, A059912, A059885, A058944, A059499, A059886-A059892.

Column k=4 of A212737.

Column k=1 gives: A218358.

Sequence in context: A289055 A109890 A086537 * A127562 A096113 A110797

Adjacent sequences:  A212483 A212484 A212485 * A212487 A212488 A212489

KEYWORD

easy,nonn,tabf

AUTHOR

Boris Putievskiy, Jun 02 2012

STATUS

approved

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Last modified October 25 16:19 EDT 2021. Contains 348255 sequences. (Running on oeis4.)