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A212737 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where column k lists the orders of degree-d irreducible polynomials over GF(prime(k)); listing order for each column: ascending d, ascending value. 12
1, 1, 3, 1, 2, 7, 1, 2, 4, 5, 1, 2, 4, 8, 15, 1, 2, 3, 3, 13, 31, 1, 2, 5, 6, 6, 26, 9, 1, 2, 3, 10, 4, 8, 5, 21, 1, 2, 4, 4, 3, 8, 12, 10, 63, 1, 2, 3, 8, 6, 4, 12, 24, 16, 127, 1, 2, 11, 6, 16, 12, 6, 16, 31, 20, 17, 1, 2, 4, 22, 9, 3, 7, 8, 24, 62, 40, 51 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
LINKS
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order
FORMULA
Formulae for the column sequences are given in A059912, A212906, ... .
EXAMPLE
For k=1 the irreducible polynomials over GF(prime(1)) = GF(2) of degree 1-4 are: x, 1+x; 1+x+x^2; 1+x+x^3, 1+x^2+x^3; 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. The orders of these polynomials p (i.e., the smallest integer e for which p divides x^e+1) are 1; 3; 7; 5, 15. (Example: (1+x^3+x^4) * (1+x^3+x^4+x^6+x^8+x^9+x^10+x^11) == x^15+1 (mod 2)). Thus column k=1 begins: 1, 3, 7, 5, 15, ... .
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
3, 2, 2, 2, 2, 2, 2, 2, 2, 2, ...
7, 4, 4, 3, 5, 3, 4, 3, 11, 4, ...
5, 8, 3, 6, 10, 4, 8, 6, 22, 7, ...
15, 13, 6, 4, 3, 6, 16, 9, 3, 14, ...
31, 26, 8, 8, 4, 12, 3, 18, 4, 28, ...
9, 5, 12, 12, 6, 7, 6, 4, 6, 3, ...
21, 10, 24, 16, 8, 8, 9, 5, 8, 5, ...
63, 16, 31, 24, 12, 14, 12, 8, 12, 6, ...
127, 20, 62, 48, 15, 21, 18, 10, 16, 8, ...
MAPLE
with(numtheory):
M:= proc(n, i) M(n, i):= divisors(ithprime(i)^n-1) minus U(n-1, i) end:
U:= proc(n, i) U(n, i):= `if`(n=0, {}, M(n, i) union U(n-1, i)) end:
b:= proc(n, i) b(n, i):= sort([M(n, i)[]])[] end:
A:= proc() local l; l:= proc() [] end;
proc(n, k) local t;
if nops(l(k))<n then l(k):= [];
for t while nops(l(k))<n
do l(k):= [l(k)[], b(t, k)] od
fi; l(k)[n]
end:
end():
seq(seq(A(n, 1+d-n), n=1..d), d=1..15);
MATHEMATICA
m[n_, i_] := Divisors[Prime[i]^n-1] ~Complement~ u[n-1, i]; u[n_, i_] := u[n, i] = If[n == 0, {}, m[n, i] ~Union~ u[n-1, i]]; b[n_, i_] := Sort[m[n, i]]; a = Module[{l}, l[_] = {}; Function[{n, k}, Module[{t}, If [Length[l[k]] < n, l[k] = {}; For[t = 1, Length[l[k]] < n, t++, l[k] = Join[l[k], b[t, k]]]]; l[k][[n]]]]]; Table[Table[a[n, 1+d-n], {n, 1, d}], {d, 1, 15}] // Flatten (* Jean-François Alcover, Dec 20 2013, translated from Maple *)
CROSSREFS
Sequence in context: A024743 A024963 A283435 * A307078 A134348 A331789
KEYWORD
nonn,look,tabl
AUTHOR
Alois P. Heinz, Jun 02 2012
STATUS
approved

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Last modified June 16 11:25 EDT 2024. Contains 373429 sequences. (Running on oeis4.)