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 1-10 of 13 results. Next

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

Views

Author

Clark Kimberling, Feb 11 2012

Keywords

Comments

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

Examples

			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

Links

Crossrefs

Cf. A206073, A206074, A257000.
Cf. also A091221, A206442.

Programs

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

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Feb 03 2012

Keywords

Comments

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

Examples

			(See the example at A206073.)

Links

Crossrefs

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).

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

Formula

Other identities and observations. For all n >= 1:
A255574(a(n)) = n.

Extensions

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

A205783 Complement of A206074, a coding of reducible polynomials over Q (with coefficients 0 or 1).

Original entry on oeis.org

1, 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, 70, 72, 74, 75, 76, 78, 80, 82, 84, 85, 86, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100
Offset: 1

Views

Author

Clark Kimberling, Feb 03 2012

Keywords

Comments

Reducibility here refers to the field of rational numbers.
Except for its initial 3, is A039004 a subsequence of A205783?

Examples

			The reducible polynomials matching the first four terms:
1 = 1(base 2) matches 1
4 = 100(base 2) matches x^2
6 = 110(base 2) matches x^2 + x
8 = 1000(base 2) matches x^3
9 = 1001(base 2) matches x^3 + 1

Links

Crossrefs

Cf. A206074 (complement), A255573 (left inverse).
After 1 a subsequence of A091212 (69 is the first term missing from here).
Cf. also permutations A260421 - A260426.

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 *)
  • PARI
    isA205783(n) = ((n > 0) && !polisirreducible(Pol(binary(n))));
n = 0; i = 0; while(n < 32768, n++; if(isA205783(n), i++; write("b205783.txt", i, " ", n)));
\\ Antti Karttunen, Jul 28 2015 after Joerg Arndt's code for A206074.

Formula

Other identities and observations. For all n >= 1:
A255573(a(n)) = n.

A206821 Numbers that match irreducible polynomials over {-1,0,1} with leading coefficient 1.

Original entry on oeis.org

2, 3, 7, 8, 10, 14, 16, 18, 21, 23, 29, 31, 35, 41, 42, 44, 48, 50, 54, 56, 60, 62, 66, 70, 72, 76, 78, 80, 82, 84, 86, 88, 93, 97, 99, 103, 109, 111, 115, 117, 123, 125, 129, 131, 137, 141, 143, 147, 153, 155, 159, 161, 165, 167, 171, 173, 179, 183, 186, 188
Offset: 1

Views

Author

Clark Kimberling, Feb 12 2012

Keywords

Comments

The monic polynomials y(n,x) having coefficients in {-1,0,1} are matched to the set N of positive integers as follows. First, the monic polynomials p(n,x) having coefficients in {0,1} are matched to N as in A206074; i.e., 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). Then monic polynomials having at least one negative coefficient are then inserted among the polynomials p(n,x), as follows: x-1 goes between x and x+1, and for k>1, the polynomials x^k-p(n,x), for 0
n ..... y(n,x) ... irreducible
1 ..... 1 ........ no
2 ..... x ........ yes
3 ..... 1+x ...... yes
4 ..... x^2 ...... no
5 .... -1+x^2 .... no
6 .... -x+x^2 .... no
7 .... -1-x+x^2 .. yes
8 ..... 1+x^2 .... yes
9 ..... x+x^2 .... no
10 .... 1+x+x^2 .. yes
11 .... x^3 ...... no
...
Guide to sequences based on the polynomials y(n,x):
A206822, irreducible
A206829, number of distinct factors
A207187, multiples of x+1
A207188, multiples of x
A207189, multiples of x-1
A207190, multiples of x^2+1
A207191, even: y(n,-x)=y(n,x)
A207192, odd: y(n,-x)=-y(n,x)

Crossrefs

Cf. A206073, A206284, A206822.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 1000}];
b[n_] := Reverse[Table[x^k, {k, 0, n}]];
p[n_] := p[n] = t[[n]].b[-1 + Length[t[[n]]]];
TableForm[Table[{n, p[n], Factor[p[n]]}, {n, 1, 6}]]
f[k_] := 2^k - k; g[k_] := 2^k - 2 + f[k - 1];
q1[n_] := p[2^(k - 1)] - p[n + 1 - f[k]];
q2[n_] := p[n - f[k] + 2];
y1 = Table[p[n], {n, 1, 4}];
Do[AppendTo[y1,Join[Table[q1[n], {n, f[k], g[k] - 1}],
   Table[q2[n], {n, g[k], f[k + 1] - 1}]]], {k, 3, 8}]
y = Flatten[y1]; (* polynomials over {-1,0,1} *)
w = {}; Do[n++; If[IrreduciblePolynomialQ[y[[n]]], AppendTo[w, n]], {n, 200}]
w                          (* A206821 *)
Complement[Range[200], w]  (* A206822 *)

A207813 Numbers that match irreducible Zeckendorf polynomials.

Original entry on oeis.org

2, 4, 9, 17, 19, 25, 27, 30, 40, 43, 46, 53, 56, 59, 61, 67, 69, 72, 77, 82, 85, 93, 95, 98, 101, 103, 108, 111, 114, 119, 124, 129, 135, 137, 140, 150, 153, 161, 166, 169, 171, 177, 179, 182, 187, 195, 197, 205, 208, 211, 213, 218, 224, 229, 237, 239
Offset: 1

Views

Author

Clark Kimberling, Feb 20 2012

Keywords

Comments

The Zeckendorf representation of a positive integer n is a unique sum
c(k-2)F(k) + c(k-3)F(k-1) + ... + c(1)F(3) + c(0)F(2),
where F=A000045 (Fibonacci numbers), c(k-2)=1, and for j=0,1,...,k-3, there are two restrictions on coefficients: c(j) is 0 or 1, and c(j)c(j+1)=0; viz., no two consecutive Fibonacci numbers appear. The Zeckendorf polynomial Z(n,x) is introduced here as
c(k-2)x^(k-2) + c(k-3)x^(k-3) + ... + c(1)x + c(0).
The name refers to irreducibility over the field of rational numbers.

Examples

			n   k    Z(n)   Z(n,x)       irreducible
1   2       1   1            no
2   3      10   x            yes
3   4     100   x^2          no
4   4     101   x^2 + 1      yes
5   5    1000   x^3          no
6   5    1001   x^3 + 1      no
7   5    1010   x^3 + x      no
8   5   10000   x^4          no
9   5   10001   x^4 + 1      yes

Crossrefs

Cf. A206073, A206074.

Programs

  • Mathematica
    fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]],
 t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k],
 AppendTo[fr, 1]; t = t - Fibonacci[k],
 AppendTo[fr, 0]]; k--]; fr]; t = Table[fb[n],
     {n, 1, 350}];
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, 40}] (* Zeckendorf polynomials *)
u = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
  AppendTo[u, n]], {n, 300}]; u     (* A207813 *)

A208179 Numbers that match polynomials with coefficients in {0,1} that have a factor containing 2 as a coefficient; see Comments.

Original entry on oeis.org

141, 177, 183, 237, 282, 354, 366, 427, 474, 555, 564, 573, 663, 669, 699, 708, 711, 717, 723, 732, 741, 753, 813, 849, 854, 871, 885, 909, 923, 933, 948, 951, 1047, 1085, 1110, 1115, 1119, 1128, 1131, 1145, 1146, 1253, 1265, 1299, 1326, 1335
Offset: 1

Views

Author

Clark Kimberling, Feb 24 2012

Keywords

Comments

The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
p(1,x) = 1
p(2,x) = x
p(3,x) = x + 1
p(4,x) = x^2
p(141,x) = x^7 + x^3 + + x^2 + 1 = (x + 1)*f(x), where
f(x) = x^6 - x^5 + x^4 - x^3 + 2*x^2 - x + 1. This shows that a factor of p(141,x) has a factor that has 2 as a coefficient. Actually, 141 is the least n for which p(n,x) has a coefficient not in {-1,0,1}.
The enumeration scheme for all nonzero polynomials with coefficients in {0,1} is introduced in Comments at A206073. The sequence A206073 itself enumerates only those polynomials that are irreducible over the ring of polynomials having integer coefficients; therefore, A206073 and A208179 are disjoint.

Examples

			The first five polynomial factors having 2 as a coefficient are indicated here:
  n ..... coefficients of a factor of p(n,x)
  141 ... 1, -1, 2, -1, 1, -1, 1  (see Comments)
  177 ... 1, -1, 1, -1, 2, -1
  183 ... 1, 0, 1, -1, 2, -1, 1
  237 ... 1, -1, 2, -1, 1, 0, 1
  282 ... 1, -1, 2, -1, 1, -1, 1  (same as for n=141)

Crossrefs

Cf. A208180, A206073, A206284, A208181, A208182.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 3000}];
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], Factor[p[n, x]]}, {n, 1, 1500}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {_, 2, _}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 1500}]]]]], {_, {}}]
Map[#[[1]] &, %]
(* Peter J. C. Moses, Feb 22 2012 *)

A208180 Numbers that match polynomials over {0,1} that have a factor containing -2 as a coefficient; see Comments.

Original entry on oeis.org

663, 669, 741, 933, 1326, 1338, 1421, 1482, 1866, 2163, 2181, 2199, 2229, 2247, 2289, 2387, 2469, 2499, 2577, 2589, 2613, 2631, 2643, 2649, 2652, 2661, 2676, 2679, 2757, 2769, 2842, 2949, 2964, 2973, 3115, 3129, 3237, 3241, 3297, 3395
Offset: 1

Views

Author

Clark Kimberling, Feb 24 2012

Keywords

Comments

The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
p(1,x) = 1
p(2,x) = x
p(3,x) = x + 1
p(4,x) = x^2
p(663,x) = 1 + x + x^2 + x^4 + x^7 + x^9 = (x + 1)*f(x), where f(x) = 1 + x^2 - x^3 + 2 x^4 - 2 x^5 + 2 x^6 - x^7 + x^8. This show that a factor of p(663,x) has a factor that has -2 as a coefficient. Actually, 663 is the least n for which p(n,x) has a coefficient not in {-1,0,1,2}.
The enumeration scheme for all nonzero polynomials with coefficients in {0,1} is introduced in Comments at A206073. The sequence A206073 itself enumerates only those polynomials that are irreducible over the ring of polynomials having integer coefficients; therefore, A206073 and A208180 are disjoint.

Crossrefs

Cf. A208179, A206073, A206284, A208181, A208182.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 4000}];
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], Factor[p[n, x]]}, {n, 1, 4000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {_, -2, _}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 3600}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208180 *)
(* Peter J. C. Moses, Feb 22 2012 *)

A208181 Numbers that match polynomials over {0,1} that have a factor containing 3 as a coefficient; see Comments.

Original entry on oeis.org

2229, 2613, 2757, 2769, 4458, 5226, 5514, 5538, 7335, 8373, 8421, 8589, 8853, 8913, 8916, 8919, 8949, 9093, 9485, 10293, 10311, 10353, 10389, 10437, 10452, 10461, 10563, 10677, 10689, 10821, 10833, 10839, 10869, 11013, 11028, 11031
Offset: 1

Views

Author

Clark Kimberling, Feb 24 2012

Keywords

Comments

The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
p(1,x) = 1
p(2,x) = x
p(3,x) = x + 1
p(4,x) = x^2
p(2229,x) =1 + x^2 + x^4 + x^5 + x^7 + x^11= (1+x)*f(x), where f(x) = 1 - x + 2 x^2 - 2 x^3 + 3 x^4 - 2 x^5 + 2 x^6 - x^7 + x^8 - x^9 + x^10. This show that a factor of p(2229,x) has a factor that has 3 as a coefficient. Actually, 2229 is the least n for which p(n,x) has a coefficient not in {-2,-1,0,1,2}.
The enumeration scheme for all nonzero polynomials with coefficients in {0,1} is introduced in Comments at A206073. The sequence A206073 itself enumerates only those polynomials that are irreducible over the ring of polynomials having integer coefficients; therefore, A206073 and A208181 are disjoint.

Crossrefs

Cf. A208179, A206073, A206284, A208180, A208182.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 15000}];
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], Factor[p[n, x]]}, {n, 1, 15000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {_, 3, _}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 14900}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208181 *)

A208182 Numbers that match polynomials over {0,1} that have a factor containing -3 as a coefficient; see Comments.

Original entry on oeis.org

8421, 8853, 9093, 10311, 10353, 10389, 10437, 10563, 10689, 10821, 10833, 10839, 10869, 11157, 12183, 12453, 14469, 14973, 14997, 16779, 16842, 17055, 17465, 17706, 18186, 18515, 18639, 19985, 20025, 20622, 20643, 20706, 20778
Offset: 1

Views

Author

Clark Kimberling, Feb 24 2012

Keywords

Comments

The polynomials having coefficients in {0,1} are enumerated at A206073. They include the following:
p(1,x) = 1
p(2,x) = x
p(3,x) = x + 1
p(4,x) = x^2
p(8421,x) = 1 + x^2 + x^5 + x^6 + x^7 + x^13
= (1 + x)*(1 + x + x^2)*f(x), where
f(x) = 1 - 2*x + 3*x^2 - 3*x^3 + 2*x^4 - x^7 + 2*x^8 - 2*x^9 + x^10.
This show that a factor of p(8421,x) has a factor that has -3 as a coefficient. Actually, 8421 is the least n for which p(n,x) has a coefficient not in {-2,-1,0,1,2,3}.
The enumeration scheme for all nonzero polynomials with coefficients in {0,1} is introduced in Comments at A206073. The sequence A206073 itself enumerates only those polynomials that are irreducible over the ring of polynomials having integer coefficients; therefore, A206073 and A208181 are disjoint.

Crossrefs

Cf. A208179, A206073, A206284, A208180, A208181.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 2], {n, 1, 25000}];
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], Factor[p[n, x]]}, {n, 1, 25000}]];
DeleteCases[
Map[{#[[1]], Cases[#[[2]], {_, -3, _}]} &,
  Map[{#[[1]], CoefficientList[#[[2]], x]} &,
   Map[{#[[1]], Map[#[[1]] &, #[[2]]]} &,
    Map[{#[[1]], Rest[FactorList[#[[2]]]]} &,
     Table[{n, Factor[p[n, x]]}, {n, 1, 24900}]]]]], {_, {}}]
Map[#[[1]] &, %]   (* A208182 *)
(* Peter J. C. Moses, Feb 22 2012 *)

A207671 Numbers in ternary representation that match nonzero polynomials with all coefficients in {0,1,2} that are irreducible modulo 3.

Original entry on oeis.org

10, 11, 12, 20, 21, 22, 101, 112, 122, 202, 211, 221, 1021, 1022, 1102, 1112, 1121, 1201, 1211, 1222, 2011, 2012, 2102, 2111, 2122, 2201, 2212, 2221, 10012, 10022, 10102, 10111, 10121, 10202, 11002, 11021, 11101, 11111, 11122, 11222
Offset: 1

Views

Author

Clark Kimberling, Feb 26 2012

Keywords

Comments

For a discussion and examples in base-10 representation, see A207670. For the analogous sequence in base 2, see A206073.

Examples

			(See the Example section of A207669.)

Crossrefs

Cf. A207669, A206073.

Programs

  • Mathematica
    t = Table[IntegerDigits[n, 3], {n, 1, 1000}];
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], Modulus -> 3],
  AppendTo[u, n]], {n, 1, 400}]
u                           (* A207669 *)
Complement[Range[200], %]   (* A207670 *)
b[n_] := FromDigits[IntegerDigits[u, 3][[n]]]
Table[b[n], {n, 1, 50}]     (* A207671 *)
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