A038569 -id:A038569 - cp's OEIS frontend

A206330 Numbers that match polynomials irreducible over the integers.

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

Clark Kimberling, Feb 06 2012

nonn

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

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

Cf. A206284 (polynomials over the positive integers),

A206331 (complement of A206330).

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 *)

A173509 Partial sums of A038568.

Original entry on oeis.org

1, 2, 4, 5, 8, 10, 13, 14, 18, 21, 25, 26, 31, 33, 38, 41, 46, 50, 55, 56, 62, 67, 73, 74, 81, 83, 90, 93, 100, 104, 111, 116, 123, 129, 136, 137, 145, 148, 156, 161, 169, 176, 184, 185, 194, 196, 205, 209, 218, 223, 232, 239, 248, 256, 265, 266, 276, 279, 289, 296

Offset: 0

Jonathan Vos Post, Feb 20 2010

Partial sums of numerators in canonical bijection from positive integers to positive rationals. The companion sequence for denominators is A173467, summing A038569. The subsequence of squares in this partial sum begins: 1, 4, 25, 81, 100, 137, 169, 196, 256, 289, 576. The subsequence of primes in this partial sum begins: 2, 5, 13, x, 41, 67, 73, 83, 223, 239, 337, 353, 379, 401, 419, 449, 479, 491, 503, 563.

			a(93) = 1 + 1 + 2 + 1 + 3 + 2 + 3 + 1 + 4 + 3 + 4 + 1 + 5 + 2 + 5 + 3 + 5 + 4 + 5 + 1 + 6 + 5 + 6 + 1 + 7 + 2 + 7 + 3 + 7 + 4 + 7 + 5 + 7 + 6 + 7 + 1 + 8 + 3 + 8 + 5 + 8 + 7 + 8 + 1 + 9 + 2 + 9 + 4 + 9 + 5 + 9 + 7 + 9 + 8 + 9 + 1 + 10 + 3 + 10 + 7 + 10 + 9 + 10 + 1 + 11 + 2 + 11 + 3 + 11 + 4 + 11 + 5 + 11 + 6 + 11 + 7 + 11 + 8 + 11 + 9 + 11 + 10 + 11 + 1 + 12 + 5 + 12 + 7 + 12 + 11 + 12 + 1 + 13 + 2.

Cf. A038568, A038569, A173467.

a(n) = SUM[i=0..n] A038568(i).

cp's OEIS Frontend

A206330 Numbers that match polynomials irreducible over the integers.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica