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A212485 Triangle T(n,k) of orders of degree-n irreducible polynomials over GF(5) listed in ascending order. 6
1, 2, 4, 3, 6, 8, 12, 24, 31, 62, 124, 13, 16, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624, 11, 22, 44, 71, 142, 284, 781, 1562, 3124, 7, 9, 14, 18, 21, 28, 36, 42, 56, 63, 72, 84, 93, 126, 168, 186, 217, 248, 252, 279, 372, 434, 504, 558, 651, 744, 868, 1116 (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 5 mod m = n, with gcd(m,5) = 1, cf. A050977.
REFERENCES
R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Cambridge Univ. Press, 1997, Table C, pp. 557-560.
LINKS
V. I. Arnol'd, Topology and statistics of formulas of arithmetics, Uspekhi Mat. Nauk, 58:4(352) (2003), 3-28
FORMULA
T(n,k) = k-th smallest element of M(n) with M(n) = {d : d | (5^n-1)} \ (M(1) U M(2) U ... U M(i-1)) for n>1, M(1) = {1,2,4}.
|M(n)| = Sum_{d|n} mu(n/d)*tau(5^d-1) = A059887.
EXAMPLE
Triangle T(n,k) begins:
1, 2, 4;
3, 6, 8, 12, 24;
31, 62, 124;
13, 16, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624;
11, 22, 44, 71, 142, 284, 781, 1562, 3124;
...
MAPLE
with(numtheory):
M:= proc(n) option remember;
`if`(n=1, {1, 2, 4}, divisors(5^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..8);
MATHEMATICA
M[n_] := M[n] = If[n == 1, {1, 2, 4}, Divisors[5^n-1] ~Complement~ U[n-1]];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n - 1]];
T[n_] := Sort[M[n]]; Array[T, 8] // Flatten (* Jean-François Alcover, Jun 10 2018, from Maple *)
CROSSREFS
Column k=3 of A212737.
Column k=1 gives: A218357.
Sequence in context: A362041 A175334 A128556 * A077661 A077583 A153125
KEYWORD
easy,nonn,look,tabf
AUTHOR
Boris Putievskiy, Jun 02 2012
STATUS
approved

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Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)