A264292
Number of irreducible polynomials in the polynomial tree T generated as in Comments.
Original entry on oeis.org
0, 0, 1, 2, 4, 7, 15, 26, 55, 101, 221, 413, 870, 1673, 3490
Offset: 0
First few generations:
g(0) = {0}
g(1) = {1}
g(2) = {2,x}
g(3) = {3, 2x, x+1, x^2}
g(4) = {4, 3x, 2x+1, 2x^2, x+2, x^2+x, x^2+1, x^3}
a(4) counts these 4 irreducible polynomials: 3x, 2x+1, x+2, x^2+1.
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z = 15; t = Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {0}, z]];
s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n]], s[n - 1]]
g[n_] := Complement[s[n], s[n - 1]]
Column[Table[g[z], {z, 1, 7}]]
Table[Count[Map[IrreduciblePolynomialQ, g[n]], True], {n, 1, z}]
A261967
{2,3,5}-primes. (See comments.)
Original entry on oeis.org
2, 151, 3061, 9517861, 11903341, 15344551, 15460771, 19975771, 37935091, 42234271, 52312411, 199938421, 228523501, 237049321, 270798991, 315266641, 315522931, 327445201, 354600601, 423223741, 466801171, 498309631, 499063711, 547916791, 585381361, 621504721
Offset: 1
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{b1, b2, b3} = {2, 3, 5}; z = 10000000;
Select[Prime[Range[z]],
PrimeQ[FromDigits[IntegerDigits[#, b1], b2]] &&
PrimeQ[FromDigits[IntegerDigits[#, b1], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b1]] &&
PrimeQ[FromDigits[IntegerDigits[#, b2], b3]] &&
PrimeQ[FromDigits[IntegerDigits[#, b3], b1]] &&
PrimeQ[FromDigits[IntegerDigits[#, b3], b2]] &]
(* Peter J. C. Moses, Sep 27 2015 *)
A264293
Number of irreducible polynomials in the n-th generation of polynomials generated as in Comments.
Original entry on oeis.org
0, 0, 2, 4, 9, 20, 54, 131, 354, 912, 2457, 6429, 17081, 44850, 118578, 311471
Offset: 0
First few generations:
g(0) = {0}
g(1) = {1}
g(2) = {2,x,y}
g(3) = {3, 2x, x^2, 1+x, 2y, xy, y^2, 1+y}
The irreducible polynomials in g(3) are 2x, 1+x, 2y, 1+y, so that a(3) = 4.
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A[0]:= 0: A[1]:= 0:
T:= {1}:
for n from 2 to 13 do
T:= map(t -> (t+1,expand(x*t),expand(y*t)),T);
A[n]:= nops(select(irreduc,T));
od:
seq(A[i],i=0..13); # Robert Israel, Nov 22 2018
-
z = 12; t = Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#, y*#} &, #], 1]] &, {0}, z]];
s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n]], s[n - 1]]
g[n_] := Complement[s[n], s[n - 1]]
Table[Length[g[z]], {z, 1, z}]
Column[Table[g[z], {z, 1, 6}]]
Table[Count[Map[IrreduciblePolynomialQ, g[n]], True], {n, 1, z}]
Showing 1-3 of 3 results.
Comments