A175198 a(n) is the smallest integer k such that the polynomial x^k + 1 over the integers mod 2 has exactly n distinct irreducible factors.

1, 3, 7, 27, 15, 21, 31, 45, 73, 91, 129, 85, 63, 93, 105, 275, 257, 219, 127, 189, 217, 453, 441, 357, 601, 741, 273, 837, 1191, 981, 513, 645, 903, 949, 255, 315, 1649, 341, 1103, 1235, 455, 651, 657, 1443, 775, 2795, 825, 1925, 1911, 771

Offset: 1

Author

Michel Lagneau, Mar 02 2010

Keywords

• nonn

Comments

Example: the polynomial x^1649 + 1 over GF(2) is the product of 37 irreducible factors.

Records: 1, 3, 7, 27, 31, 45, 73, 91, 129, 275, 453, 601, 741, 837, 1191, 1649, 2795, 3045, 3913, 3955, 10719, 18875, 48631, 73143, 76373, 126191, 189061, 210105, 216481, 249891, 303021, 896041, 961185, 1063997, 1759603, 2555521, 3492783, 3923381, 5276409, 5529727, 6663515, 7234645, 8761553, 10488401, 11636993, 12290949, 20936365, 25099273, 25821285, 28081875, 28623469, 32848947, 48539883, 58885551, ..., . - Robert G. Wilson v, Feb 07 2018

Examples

			For n=1, x+1 is irreducible hence a(1) = 1.
For n=2, x^3 + 1 = (x+1)(x^2+x+1) mod 2 hence a(2) = 3.
For n=3, x^7 + 1 = (x+1)(x^3 + x^2 + 1)(x^3 + x + 1) mod 2 hence a(3) = 7.
For n=4, x^27 + 1 = (x+1)(x^2 + x + 1)(x^18 + x^9 + 1)(x^6 + x^3 + 1) mod 2 hence a(4) = 27.
For n=5, x^15 + 1 = (x+1)(x^4 + x^3 + x^2 + x + 1)(x^2 + x + 1)(x^4 + x^3 + 1)(x^4 + x + 1) mod 2 hence a(5) = 15.
		

References

• R. Lidl and H. Niederreiter, Finite Fields, Addison-Wesley, 1983, p. 65.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 300 terms from Charles R Greathouse IV)
• Eric Weisstein's World on Math, Irreducible Polynomial

Crossrefs

Cf. A023135, A000005, A000374, A023136-A023142.

Programs

Maple

with(numtheory):T:=array(0..50000000):U=array(0..50000000 ):nn:=3000: for k from 1 to nn do:liste:=Factors(x^k+ 1) mod 2;t1 := liste[2]:t2:=(liste[2][i], i=1..nops(t1)):a :=nops(t1):T[k]:=a:U[k]:=k:od:mini:=T[1]:ii:=1: print(mini):for p from 1 to nn-1 do:for n from 1 to nn-1 do:if T[n] < mini then mini:= T[n]:ii:=n: indice:=U[n]: else fi:od:print(indice):print(mini):T[ii]:= 99999999: ii:=1:mini:=T[1] :od:

Mathematica

With[{s = Array[Length@ FactorList[#, Modulus -> 2] &[x^# + 1] &, 500]}, Array[FirstPosition[s, #][[1]] &, 1 + LengthWhile[Differences@ #, # == 1 &], 2] &@ Union@ s] (* Michael De Vlieger, Feb 05 2018 *)
CountFactors[p_, n_] := CountFactors[p, n] = Module[{sum = 0, m = n, d, f, i}, While[ Mod[m, p] == 0, m /= p]; d = Divisors[m]; Do[f = d[[i]]; sum += EulerPhi[f]/MultiplicativeOrder[p, f], {i, Length[d]}]; sum](*after Shel Kaphan from A000374*); f[n_] := Block[{k = 1}, While[CountFactors[2, k] != n, k++]; k]; Array[f, 60] (* Robert G. Wilson v, Feb 07 2018 *)

PARI

first(n)=my(v=vector(n),left=n,k,t); while(left, t=#factor(Mod('x^k+++1,2))~; if(t<=n && v[t]==0, v[t]=k; left--)); v \\ Charles R Greathouse IV, Jan 28 2018

Formula

A000005(a(n)) <= n <= a(n). - Robert Israel, Feb 07 2018

Extensions

Name corrected by Charles R Greathouse IV, Jan 28 2018