A091221 -id:A091221 - cp's OEIS frontend

A304108 Numbers n such that the (0,1)-polynomial encoded in binary expansion of n has at least one duplicated irreducible divisor when the factorization is done in polynomial ring GF(2)[X].

4, 5, 8, 10, 12, 15, 16, 17, 20, 21, 24, 27, 28, 30, 32, 34, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 65, 68, 69, 72, 75, 76, 78, 80, 81, 84, 85, 88, 90, 92, 95, 96, 99, 100, 102, 104, 105, 107, 108, 112, 114, 116, 119, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 141, 144, 147, 148, 150, 151, 152, 153, 156, 160, 162

Offset: 1

Antti Karttunen, May 13 2018

nonn

Positions of zeros in A091219 and A304109. Numbers n such that A091221(n) < A091222(n).

			4 is present as 4 = A048720(2,2) = A048720(A014580(1), A014580(1)).
5 is present as 5 = A048720(3,3) = A048720(A014580(2), A014580(2)).
10 is present as 10 = A048720(2,A048720(3,3)).

Antti Karttunen, Table of n, a(n) for n = 1..32768
Index entries for sequences related to polynomials in ring GF(2)[X]

Cf. A304107 (complement).
Cf. A014580, A048720, A091219, A091221, A091222, A304109.
Cf. also A013929.

isA304108(n) = { my(fm=factor(Pol(binary(n))*Mod(1, 2))); for(k=1, #fm~, if(fm[k, 2] > 1, return(1))); (0); };
k=0; n=0; while(k<100, n++; if(isA304108(n), k++; print1(n,", ")));

A206719 Number of distinct irreducible factors of the polynomial p(n,x) defined at A206073.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 2, 3, 2, 1, 2, 2, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 3, 1, 2, 3, 1, 2, 1, 3, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 1, 2, 1, 3, 2, 2, 1, 2, 1, 4, 1, 2, 2, 2, 2, 2, 1, 3, 2

Offset: 1

Clark Kimberling, Feb 11 2012

nonn

The polynomials having coefficients in {0,1} are enumerated as in A206074 (and A206073).

			p(1,n) = 1, so a(1)=0
p(2,n) = x, so a(2)=1
p(6,n) = x(1+x), so a(6)=2
p(18,n) = x(x+1)(1-x+x^2), so a(18)=3
p(90,n) = x(1+x)(1+x^2)(1-x+x^2), so a(90)=4

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A206073, A206074, A257000.

Cf. also A091221, A206442.

t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := p[n, x] = t[[n]].b[-1 + Length[t[[n]]]]
TableForm[Table[{n, p[n, x],
   FactorList[p[n, x]], -1 + Length[FactorList[p[n, x]]]}, {n, 1, 9}]]
Table[Length[FactorList[p[n, x]]], {n, 1, 120}]

A206719(n) = { my(f = factor(Pol(binary(n)))); (#f~); }; \\ Antti Karttunen, Dec 16 2017

A136378 Number of distinct irreducible polynomials dividing A036284(n), when it is considered as a GF(2)[X]-polynomial.

Original entry on oeis.org

2, 2, 3, 4, 5, 5, 5, 7, 7, 10, 15, 10, 10, 12, 15, 11

Offset: 0

Antti Karttunen, Dec 29 2007

nonn

a(n) = A091221(A036284(n)).

A376932 a(n) is the index of the first GF(2)[X] polynomial that has n distinct irreducible factors.

Original entry on oeis.org

1, 2, 6, 18, 166, 1806, 20382, 272706, 8323326, 158143194, 4319806194, 139715547110, 4563596609414, 154716297384250, 6051527318503338, 315946019303255670, 18477283150919171654, 1191953715632050834242, 76457609628854745786262, 4838004466153152832995822, 312401901306255000752991994, 20039165126917559409941672886

Offset: 0

Robert Israel, Oct 11 2024

nonn

a(n) is the least k such that A091221(k) = n.

			a(3) = 18 because the 18th GF(2)[X] polynomial is X^4 + X = X * (X + 1) * (X^2 + X + 1) with 3 distinct irreducible factors over GF(2).

Cf. A091221.

pol:= proc(x) local L,i;
  L:= convert(x,base,2);
  add(L[i]*X^(i-1),i=1..nops(L));
end proc:
for m from 1 to 10 do
  IP[m]:= select(t -> Irreduc(pol(t)) mod 2, [seq(x,x=2^m..2^(m+1)-1)]);
od:
nIP:= [seq(nops(IP[m]),m=1..10)]:
psnIP:= ListTools:-PartialSums(nIP):
f:= proc(n) local k,P0,r, xmin, x, i, s, P;
  for k from 1 while n > psnIP[k] do od:
  P0:= expand(mul(convert(map(pol,IP[i]),`*`),i=1..k-1)) mod 2;
  if k = 1 then r:= n else r:= n - psnIP[k-1] fi;
  xmin:= infinity;
  for s in combinat:-choose(IP[k],r) do
    P:= expand(P0 * mul(pol(i),i=s)) mod 2;
    x:= eval(P,X=2);
    xmin:= min(xmin, x);
  od;
  xmin
end proc:
seq(f(i),i=0..25);

cp's OEIS Frontend

A304108 Numbers n such that the (0,1)-polynomial encoded in binary expansion of n has at least one duplicated irreducible divisor when the factorization is done in polynomial ring GF(2)[X].

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

A206719 Number of distinct irreducible factors of the polynomial p(n,x) defined at A206073.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A136378 Number of distinct irreducible polynomials dividing A036284(n), when it is considered as a GF(2)[X]-polynomial.

Views

Author

Keywords

Links

Crossrefs

A376932 a(n) is the index of the first GF(2)[X] polynomial that has n distinct irreducible factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple