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.

A206285 Numbers that match polynomials not irreducible over the nonnegative integers.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 11, 13, 14, 15, 16, 17, 19, 21, 23, 25, 26, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 69, 71, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94, 95, 97, 99, 101, 103
Offset: 1

Views

Author

Clark Kimberling, Feb 05 2012

Keywords

Comments

Complement of A206284.

Examples

			(See the example at A206284.)

Crossrefs

Cf. A206285, A206331.

Programs

  • Mathematica
    b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 400;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
== Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
    Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
      Length[f[n]]}]], {n, 1, z}];
p[n_, x_] := u[[n]].b[-1 + Length[u[[n]]]]
Table[p[n, x], {n, 1, z/4}]
v = {}; Do[n++; If[IrreduciblePolynomialQ[p[n, x]],
AppendTo[v, n]], {n, z/2}]; v  (* A206284 *)
Complement[Range[200], v]      (* A206285 *)

A206330 Numbers that match polynomials irreducible over the integers.

Original entry on oeis.org

3, 4, 5, 6, 9, 10, 17, 18, 19, 20, 21, 22, 29, 30, 33, 34, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 53, 54, 55, 56, 57, 58, 59, 60, 69, 70, 73, 74, 77, 78, 81, 82, 83, 84, 87, 88, 97, 98, 101, 102, 105, 106, 109, 110, 113, 114, 117, 118, 119, 120, 123
Offset: 1

Views

Author

Clark Kimberling, Feb 06 2012

Keywords

Comments

Each n>1 matches a polynomial having integer coefficients
determined by the prime factorization of n. Let c be a
positive integer, and write
c=p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k), and
define p(n,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.
If c/d is a rational number with GCD(c,d)=1, define
Q(c/d,x)=p(c,x)-p(d,x). Let c(n)/d(n) be the n-th
positive rational number given by the canonical
bijection; i.e., c(n)=A038568(n)/A038569(n).
Define P(0,x)=1 and P(n,x)=Q(c(n)/d(n),x). Polynomials
having nonnegative integer coefficients are matched to
the nonnegative integers as follows:
...
n .... P[n,x] .. irreducible
0 .... 0 ....... no
1 ... -1 ....... no
2 .... 1 ....... no
3 ... -x ....... yes
4 .... x ....... yes
5 ... 1-x ...... yes
6 .. -1+x ...... yes
7 .. -2 ........ no
8 ... 2 ........ no
9 .. -2+x ...... yes
10 .. 2-x ...... yes

Examples

			In the table under Comments, read "yes" for n=3,4,5,6,9,10.

Crossrefs

Cf. A206284 (polynomials over the positive integers),
A206331 (complement of A206330).

Programs

  • Mathematica
    b[n_] := Table[x^k, {k, 0, n}];
f[n_] := f[n] = FactorInteger[n]; z = 1000;
t[n_, m_, k_] := If[PrimeQ[f[n][[m, 1]]] && f[n][[m, 1]]
 == Prime[k], f[n][[m, 2]], 0];
u = Table[Apply[Plus,
    Table[Table[t[n, m, k], {k, 1, PrimePi[n]}], {m, 1,
      Length[f[n]]}]], {n, 1, z}];
c[n_] := Module[{s = 1, k = 2, j = 1},
   While[s <= n, s = s + 2*EulerPhi[k]; k = k + 1];
   s = s - 2*EulerPhi[k - 1];
   While[s <= n, If[GCD[j, k - 1]
      == 1, s = s + 2]; j = j + 1];
   If[s > n + 1, j - 1, k - 1]];
d[n_] := Module[{s = 1, k = 2, j = 1},
   While[s <= n, s = s + 2*EulerPhi[k]; k = k + 1];
   s = s - 2*EulerPhi[k - 1];
   While[s <= n, If[GCD[j, k - 1]
      == 1, s = s + 2]; j = j + 1];
   If[s > n + 1, k - 1, j - 1]];
P[n_, x_] :=
 u[[c[n]]].b[-1 + Length[u[[c[n]]]]] -
  u[[d[n]]].b[-1 + Length[u[[d[n]]]]]
TableForm[Table[{n, P[n, x], Factor[P[n, x]]},
   {n, 1, z/4}]];
v = {}; Do[n++;
 If[IrreduciblePolynomialQ[P[n, x]], AppendTo[v, n]], {n, z/2}]
v                            (* A206330 *)
Complement[Range[0,200], v]  (* A206331 *)

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

Showing 1-3 of 3 results.

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