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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
Eric Weisstein's World of Mathematics, Irreducible Polynomial
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);
MATHEMATICA
M[n_] := M[n] = If[n == 1, {1, 2, 3, 6}, Divisors[7^n - 1] ~Complement~ U[n - 1]];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]];
Table[T[n], {n, 1, 7}] // Flatten (* Jean-François Alcover, Sep 24 2022, from Maple code *)
CROSSREFS
Column k=4 of A212737.
Column k=1 gives: A218358.
Sequence in context: A109890 A370046 A086537 * A127562 A096113 A110797
KEYWORD
easy,nonn,tabf
AUTHOR
Boris Putievskiy, Jun 02 2012
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)