A260427 -id:A260427 - cp's OEIS frontend

A206074 n-th irreducible polynomial over Q (with coefficients 0 or 1) evaluated at x=2.

2, 3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 69, 71, 73, 77, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 113, 115, 117, 121, 127, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 171, 173, 179, 181, 185, 191, 193, 197, 199, 203, 205, 209, 211, 213, 223, 227, 229

Offset: 1

Clark Kimberling, Feb 03 2012

nonn

Is every prime present?
Yes, see the Filaseta reference. - Thomas Ordowski, Feb 19 2014
Corresponding evaluation at x=10 is A206073. - Michael Somos, Feb 26 2014

			(See the example at A206073.)

Antti Karttunen, Table of n, a(n) for n = 1..21692
John Brillhart, Michael Filaseta, Andrew Odlyzko, On an irreducibility theorem of A. Cohn, Canad. J. Math. 33(1981), pp. 1055-1059.
Michael Filaseta, A further generalization of an irreducibility theorem of A. Cohn, Canad J. Math. 34 (1982), pp. 1390-1395.

Cf. A206073, A205783 (complement), A206075 (nonprime terms), A014580 (irreducible over GF(2), a subsequence of this one), A000040 (primes, also a subsequence), A260427 (terms that are reducible over GF(2)).
Cf. A255574 (left inverse).
Cf. also permutations A260421 - A260426.
Disjoint union of A257688 (without 1) and A260428.
a(n) differs from A186891(n+1) for the first time at n=21, where a(21) = 67, while A186891(22) = 65, a term missing from here. There are several other sequences that do not diverge until after approx. the twentieth term from this one (see the context-links).

t = Table[IntegerDigits[n, 2], {n, 1, 850}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]]
p[n_, x_] := t[[n]].b[-1 + Length[t[[n]]]]
Table[p[n, x], {n, 1, 15}]
u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
AppendTo[u, n]], {n, 300}];
u                          (* A206074 *)
Complement[Range[200], u]  (* A205783 *)
b[n_] := FromDigits[IntegerDigits[u, 2][[n]]]
Table[b[n], {n, 1, 40}]    (* A206073 *)

for(n=2, 10^3, if( polisirreducible( Pol(binary(n)) ), print1(n,", ") ) ); \\ Joerg Arndt, Feb 19 2014

Other identities and observations. For all n >= 1:

A255574(a(n)) = n.

Clarified name, added more terms, Joerg Arndt, Feb 20 2014

A260426 a(1) = 1, a(A206074(n)) = A014580(a(n)), a(A205783(1+n)) = A091242(a(n)), where A014580 [respectively A091242] give binary codes for irreducible [resp. reducible] polynomials over GF(2), while A206074 and A205783 give similar codes for polynomials with coefficients 0 or 1 that are irreducible [resp. reducible] over Q.

Original entry on oeis.org

1, 2, 3, 4, 7, 5, 11, 6, 8, 12, 25, 9, 13, 17, 10, 14, 47, 18, 19, 34, 15, 20, 31, 24, 55, 16, 21, 62, 137, 26, 37, 27, 45, 22, 28, 42, 59, 33, 71, 23, 87, 29, 41, 79, 166, 35, 61, 49, 36, 58, 30, 38, 319, 54, 91, 76, 44, 89, 97, 32, 203, 108, 39, 53, 99, 200, 67, 46, 103, 78, 185, 64, 131, 48, 75, 40, 379, 50, 73, 373, 109, 70, 433, 113, 95, 57, 1123, 111, 143, 121

Offset: 1

Antti Karttunen, Jul 26 2015

Each term of A260427 resides in a separate infinite cycle. This follows because any polynomial with (coefficients 0 or 1) that is irreducible over GF(2) is also irreducible over Q, in other words, A014580 is a subset of A206074. [See Thomas Ordowski's Feb 21 2014 comment in A014580] and thus any term of A091242 in A206074 is trapped into a trajectory containing only terms of A014580.

Antti Karttunen, Table of n, a(n) for n = 1..7968
Index entries for sequences that are permutations of the natural numbers

Inverse: A260425.
Related permutations: A246202, A245703, A260421, A260424.
Cf. A014580, A091242, A205783, A206074, A257000, A255572, A255574, A260427.
Differs from A245703 for the first time at n=25, where a(25)=55, while A245703(25)=16.

allocatemem(234567890);
vecsize = (2^24)-4;
uplim = 2^25;
v014580 = vector(vecsize); A014580 = n -> v014580[n];
v091242 = vector(vecsize); A091242 = n -> v091242[n];
v255574 = vector(vecsize); A255574 = n -> v255574[n];
A255572 = n -> (n - A255574(n) - 1);
isA014580(n) = polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA206074(n) = polisirreducible(Pol(binary(n)));
A257000 = n -> isA206074(n);
v255574[1] = 0; i=0; j=0; n=2; while((n < uplim), if(!(n%65536),print1(n,", ")); v255574[n] = v255574[n-1]+A257000(n); if(isA014580(n), i++; v014580[i] = n, j++; v091242[j] = n); n++); print(n);
A260426(n) = if(1==n, 1, if(isA206074(n), A014580(A260426(A255574(n))), A091242(A260426(A255572(n)))));
for(n=1, 7968, write("b260426.txt", n, " ", A260426(n)));

a(1) = 1; for n > 1, if A257000(n) = 1 [when n is in A206074], then a(n) = A014580(a(A255574(n))), otherwise [when n is in A205783], a(n) = A091242(a(A255572(n))).
As a composition of related permutations:
a(n) = A246202(A260421(n)).
a(n) = A245703(A260424(n)).

A260428 Composite numbers whose binary representations encode a polynomial (with coefficients 0 or 1) which is irreducible over Q, but reducible over GF(2).

Original entry on oeis.org

69, 77, 81, 121, 169, 205, 209, 261, 265, 275, 289, 295, 305, 321, 323, 327, 329, 339, 377, 405, 407, 437, 453, 473, 475, 481, 493, 517, 533, 551, 553, 559, 565, 575, 581, 583, 595, 625, 649, 667, 671, 689, 703, 707, 737, 747, 749, 755, 763, 767, 779, 781, 785, 805, 815, 833, 835, 851, 855, 861, 869, 893, 905

Offset: 1

Antti Karttunen, Jul 26 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..11585

Intersection of A002808 and A260427.
Intersection of A091212 and A206074.
Intersection of A091242 and A206075.
Complement of A257688 in A206074.

f:= proc(n) local L,p,x;
  if isprime(n) then return false fi;
  L:= convert(n,base,2);
  p:= add(L[i]*x^(i-1),i=1..nops(L));
  irreduc(p) and not (Irreduc(p) mod 2);
end proc:
select(f, [$2..10000]); # Robert Israel, Jul 27 2015

okQ[n_] := CompositeQ[n] && Module[{id, pol, x}, id = IntegerDigits[n, 2] // Reverse; pol = id.x^Range[0, Length[id]-1]; IrreduciblePolynomialQ[pol] && !IrreduciblePolynomialQ[pol, Modulus -> 2]];
Select[Range[1000], okQ] (* Jean-François Alcover, Feb 06 2023 *)

isA260428(n) = (polisirreducible( Pol(binary(n)) ) && !polisirreducible(Pol(binary(n))*Mod(1, 2)) && !isprime(n));
n = 0; i = 0; while(n < 65537, n++; if(isA260428(n), i++; write("b260428.txt", i, " ", n)));

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A206074 n-th irreducible polynomial over Q (with coefficients 0 or 1) evaluated at x=2.

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A260428 Composite numbers whose binary representations encode a polynomial (with coefficients 0 or 1) which is irreducible over Q, but reducible over GF(2).

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