cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A091206 Primes whose binary representation encodes a polynomial irreducible over GF(2).

Original entry on oeis.org

2, 3, 7, 11, 13, 19, 31, 37, 41, 47, 59, 61, 67, 73, 97, 103, 109, 131, 137, 157, 167, 191, 193, 211, 229, 239, 241, 283, 313, 379, 397, 419, 433, 463, 487, 499, 557, 563, 587, 601, 607, 613, 617, 631, 647, 661, 677, 701, 719, 757, 761, 769, 787, 827, 859
Offset: 1

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Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

"Encoded in binary representation" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where each coefficient a(k) = 0 or 1).
Subsequence with Hamming weight nonprime starts 2, 1019, 1279, 1531, 1663, 1759, 1783, 1789, 2011, 2027, 2543, 2551, ... [Joerg Arndt, Nov 01 2013]. These are now given by A255569. - Antti Karttunen, May 14 2015

Links

Crossrefs

Intersection of A014580 and A000040.
Apart from a(2) = 3 a subsequence of A027697. The numbers in A027697 but not here are listed in A238186.
Also subsequence of A235045 (its primes. Cf. also A235041-A235042).
Cf. A091209 (Primes whose binary expansion encodes a polynomial reducible over GF(2)), A091212 (Composite, and reducible over GF(2)), A091214 (Composite, but irreducible over GF(2)), A257688 (either 1, prime or irreducible over GF(2)).
Subsequence: A255569.

Programs

  • Mathematica
    okQ[p_] := Module[{id, pol, x}, id = IntegerDigits[p, 2] // Reverse; pol = id.x^Range[0, Length[id] - 1]; IrreduciblePolynomialQ[pol, Modulus -> 2]];
Select[Prime[Range[1000]], okQ] (* Jean-François Alcover, Feb 06 2023 *)
  • PARI
    is(n)=polisirreducible( Mod(1,2) * Pol(digits(n,2)) );
forprime(n=2,10^3,if (is(n), print1(n,", ")));
\\ Joerg Arndt, Nov 01 2013

Formula

a(n) = A000040(A091207(n)) = A014580(A091208(n)).

A091209 Primes whose binary representation encodes a polynomial reducible over GF(2).

Original entry on oeis.org

5, 17, 23, 29, 43, 53, 71, 79, 83, 89, 101, 107, 113, 127, 139, 149, 151, 163, 173, 179, 181, 197, 199, 223, 227, 233, 251, 257, 263, 269, 271, 277, 281, 293, 307, 311, 317, 331, 337, 347, 349, 353, 359, 367, 373, 383, 389, 401, 409, 421, 431, 439, 443, 449, 457, 461, 467, 479, 491, 503, 509, 521, 523
Offset: 1

Views

Author

Antti Karttunen, Jan 03 2004

Keywords

Comments

"Encoded in binary representation" means that a polynomial a(n)*X^n+...+a(0)*X^0 over GF(2) is represented by the binary number a(n)*2^n+...+a(0)*2^0 in Z (where each coefficient a(k) = 0 or 1).
Except for 3, all primes with even Hamming weight (A027699) are terms, see A238186 for the subsequence of primes with odd Hamming weight. [Joerg Arndt and Antti Karttunen, Feb 19 2014]

Links

Crossrefs

Intersection of A000040 and A091242.
Disjoint union of A238186 and (A027699 \ {3}).
Left inverse: A235043.
Cf. A091206 (Primes whose binary expansion encodes a polynomial irreducible over GF(2)), A091212 (Composite, and reducible over GF(2)), A091214 (Composite, but irreducible over GF(2)).
Cf. also A235041-A235042, A234742.

Programs

  • Maple
    Primes:= select(isprime,[2,seq(2*i+1,i=1..1000)]):
filter:= proc(n) local L,x;
    L:= convert(n,base,2);
    Irreduc(add(L[i]*x^(i-1),i=1..nops(L))) mod 2;
end proc:
remove(filter,Primes); # Robert Israel, May 17 2015
  • Mathematica
    Select[Prime[Range[2, 100]], !IrreduciblePolynomialQ[bb = IntegerDigits[#, 2]; Sum[bb[[k]] x^(k-1), {k, 1, Length[bb]}], Modulus -> 2]&] (* Jean-François Alcover, Feb 28 2016 *)
  • PARI
    forprime(p=2, 10^3, if( ! polisirreducible( Mod(1,2)*Pol(binary(p)) ), print1(p,", ") ) ); \\ Joerg Arndt, Feb 19 2014

Formula

a(n) = A000040(A091210(n)) = A091242(A091211(n)).
Other identities. For all n >= 1:
A235043(a(n)) = n. [A235043 works as a left inverse of this sequence.]

A246158 Odious reducible polynomials over GF(2), coded in binary. (Polynomials with an odd number of nonzero terms that are reducible over GF(2)).

Original entry on oeis.org

4, 8, 14, 16, 21, 22, 26, 28, 32, 35, 38, 42, 44, 49, 50, 52, 56, 62, 64, 69, 70, 74, 76, 79, 81, 82, 84, 88, 93, 94, 98, 100, 104, 107, 110, 112, 118, 121, 122, 124, 127, 128, 133, 134, 138, 140, 146, 148, 151, 152, 155, 158, 161, 162, 164, 168, 173, 174, 176, 179, 181, 182, 186, 188, 194, 196, 199, 200
Offset: 1

Views

Author

Antti Karttunen, Aug 20 2014

Keywords

Comments

Self-inverse permutation A193231 maps each term of this sequence to some term of A246156 and vice versa.
Each term belongs into a distinct infinite cycle in permutations like A246161/A246162 and A246163/A246164 apart from 4, which is in a finite cycle (3 4) of A246161/A246162 and 4 and 8 which both are in the same (infinite) cycle of A246163/A246164.

Examples

			4, which is 100 in binary, encodes polynomial x^2, which factorizes as (x)(x) over GF(2), (4 = A048720(2,2)), thus it is reducible in that polynomial ring. It also has an odd number of nonzero terms present (equally: odd number of 1-bits in its code), in this case just one, thus 4 is a member of this sequence.

Links

Crossrefs

Intersection of A091242 and A000069 (odious numbers).
A238186 and A246157 are subsequences.
Cf. also A048720, A246156, A193231, A246161, A246162, A246163, A246164.
Showing 1-3 of 3 results.

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