A260428 -id:A260428 - 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

A260427 Binary codes for polynomials (with coefficients 0 or 1) that are irreducible over Q, but reducible over GF(2).

Original entry on oeis.org

5, 17, 23, 29, 43, 53, 69, 71, 77, 79, 81, 83, 89, 101, 107, 113, 121, 127, 139, 149, 151, 163, 169, 173, 179, 181, 197, 199, 205, 209, 223, 227, 233, 251, 257, 261, 263, 265, 269, 271, 275, 277, 281, 289, 293, 295, 305, 307, 311, 317, 321, 323, 327, 329, 331, 337, 339, 347, 349, 353, 359, 367, 373, 377, 383, 389, 401

Offset: 1

Antti Karttunen, Jul 26 2015

nonn

Antti Karttunen, Table of n, a(n) for n = 1..16972

Intersection of A091242 and A206074.
Subsequence: A260428.
Cf. also A260426, A206075.

okQ[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 *)

isA260427(n) = (polisirreducible( Pol(binary(n)) ) && !polisirreducible(Pol(binary(n))*Mod(1, 2)));
n = 0; i = 0; while(n < 65537, n++; if(isA260427(n), i++; write("b260427.txt", i, " ", n)));

A325386 Numbers n such that for any divisor d of n and some k, A048720(d,k) = n only for trivial cases d=1 and d=n, despite that n is neither prime nor in A014580.

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, 555, 559, 565, 575, 581, 583, 595, 625, 649, 667, 671, 699, 703, 707, 737, 747, 749, 755, 763, 767, 779, 785, 805, 815, 833, 835, 849, 851, 855, 861, 869, 871, 885, 893, 905, 923, 925

Offset: 1

Antti Karttunen, May 11 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences related to polynomials in ring GF(2)[X]

Terms of A325559 not in A257688.
Subsequence of A005408 (odd numbers).
Cf. A014580, A325560.
Differs from A260428 for the first time at n=32, where a(32) = 555, a value missing from A260428.

A325560(n) = { my(p = Pol(binary(n))*Mod(1, 2)); sumdiv(n,d,my(q = Pol(binary(d))*Mod(1, 2)); !(p%q)); };
isA325386(n) = (!isprime(n) && !polisirreducible(Pol(binary(n))*Mod(1,2)) && (2 == A325560(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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A260427 Binary codes for polynomials (with coefficients 0 or 1) that are irreducible over Q, but reducible over GF(2).

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A325386 Numbers n such that for any divisor d of n and some k, A048720(d,k) = n only for trivial cases d=1 and d=n, despite that n is neither prime nor in A014580.

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