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.

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A206073 Binary numbers that represent irreducible polynomials over the rationals with coefficients restricted to {0,1}.

Original entry on oeis.org

10, 11, 101, 111, 1011, 1101, 10001, 10011, 10111, 11001, 11101, 11111, 100101, 101001, 101011, 101111, 110101, 110111, 111011, 111101, 1000011, 1000101, 1000111, 1001001, 1001101, 1001111, 1010001, 1010011, 1010111, 1011001
Offset: 1

Views

Author

Clark Kimberling, Feb 03 2012

Keywords

Comments

The polynomial x^d(0) + x^d(1) + ... + d(n), where d(i) is 0 or 1 for 0<=i<=n and d(0)=1, matches the binary number d(0)d(1)...d(n). (This is an enumeration of all the nonzero polynomials with coefficients in {0,1}, not just those that are irreducible.)

Examples

			The matching of binary numbers to the first six polynomials irreducible over the field of rational numbers:
10 .... x
11 .... x + 1
101 ... x^2 + 1
111 ... x^2 + x + 1
1011 .. x^3 + x + 1

Crossrefs

Cf. A171000 (irreducible Boolean polynomials).
Cf. A205783 (complement), A206074 (base 10).

Programs

  • Mathematica
    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 *)

A091212 Composite numbers whose binary representation encodes a polynomial reducible over GF(2).

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99
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).
From Reinhard Zumkeller, Jul 05-12 2011, values for maximum n corrected by Antti Karttunen, May 18 2015: (Start)
a(n) = A192506(n) for n <= 36.
a(n) = A175526(n) for n <= 36.
(End)

Links

Crossrefs

Intersection of A002808 and A091242.
Cf. A257688 (complement, either 1, irreducible in GF(2)[X] or prime), A091206 (prime and irreducible), A091209 (prime and reducible), A091214 (nonprime and irreducible).
Cf. A091213, A236861, A235036 (a subsequence, apart from 1).
Differs from both A175526 and A192506 for the first time at n=37, where a(37) = 56, while A175526(37) = A192506(37) = 55, a term missing from here (as 55 encodes a polynomial which is irreducible in GF(2)[X]).
Differs from its subsequence A205783(n+1) for the first time at n=47, where a(47) = 69, while 69 is missing from A205783.

Programs

  • PARI
    isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
isA091212(n) = ((n > 1) && !isprime(n) && !isA014580(n));
n = 0; i = 0; while(n < 2^16, n++; if(isA091212(n), i++; write("b091212.txt", i, " ", n)));

Formula

a(n) = A091242(A091213(n)).

A192506 Numbers that are neither ludic nor prime.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 92, 93, 94
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 05 2011

Keywords

Comments

Intersection of A002808 and A192607; (1-A010051(a(n)))*(1-A192490(a(n)))=1;
a(n) = A091212(n) for n <= 60.
a(n) = A175526(n) for n <= 53. - Reinhard Zumkeller, Jul 12 2011
In other words, composite numbers that are nonludic. - Antti Karttunen, May 11 2015

Links

Crossrefs

Cf. A257689 (complement, either ludic or prime), A192503 (ludic and prime), A192504 (ludic and nonprime), A192505 (nonludic and prime).
Cf. A010051, A192490, A002808, A192607.
a(n) differs from A091212(n) and A205783(n+1) for the first time at n=37, where a(37) = 55, while 55 is missing from both A091212 and A205783.
Differs from A175526 for the first time at n=54, where a(54) = 78, while A175526(54) = 77, a term which is missing from here.

Programs

  • Haskell
    a192506 n = a192506_list !! (n-1)
a192506_list = filter ((== 0) . a010051) a192607_list
(Scheme, with Antti Karttunen's IntSeq-library)
(define A192506 (MATCHING-POS 1 1 (lambda (n) (and (zero? (A192490 n)) (zero? (A010051 n))))))
;; Antti Karttunen, May 07 2015
  • Mathematica
    a3309[nmax_] := a3309[nmax] = Module[{t = Range[2, nmax], k, r = {1}}, While[Length[t] > 0, k = First[t]; AppendTo[r, k]; t = Drop[t, {1, -1, k}]]; r];
ludicQ[n_, nmax_] /; 1 <= n <= nmax := MemberQ[a3309[nmax], n];
terms = 1000;
f[nmax_] := f[nmax] = Select[Range[nmax], !ludicQ[#, nmax] && !PrimeQ[#]&] // PadRight[#, terms]&;
f[nmax = terms];
f[nmax = 2 nmax];
While[f[nmax] != f[nmax/2], nmax = 2 nmax];
seq = f[nmax] (* Jean-François Alcover, Dec 10 2021, after Ray Chandler in A003309 *)

A206822 Numbers that match non-irreducible monic polynomials having coefficients in {-1,0,1}; complement of A206821.

Original entry on oeis.org

1, 4, 5, 6, 9, 11, 12, 13, 15, 17, 19, 20, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 36, 37, 38, 39, 40, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 77, 79, 81, 83, 85, 87, 89, 90, 91, 92, 94, 95, 96, 98, 100, 101, 102
Offset: 1

Views

Author

Clark Kimberling, Feb 12 2012

Keywords

Comments

See A206821.

Examples

			(See A206821.)

Crossrefs

Cf. A206821, A205783, A206331.

Programs

Previous Showing 11-14 of 14 results.

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