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A206284 Numbers that match irreducible polynomials over the nonnegative integers. 23
3, 6, 9, 10, 12, 18, 20, 22, 24, 27, 28, 30, 36, 40, 42, 44, 46, 48, 50, 52, 54, 56, 60, 66, 68, 70, 72, 76, 80, 81, 88, 92, 96, 98, 100, 102, 104, 108, 112, 114, 116, 118, 120, 124, 126, 130, 132, 136, 140, 144, 148, 150, 152, 154, 160, 162, 164, 168, 170 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

After p(1) which encodes 0-polynomial, each n > 1 corresponds with a polynomial having nonnegative integer coefficients determined by the prime factorization of n.

Write n = p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k). The matching polynomial is then p(n,x) = e(1) + e(2)x + e(3)x^2 + ... + e(k)x^k.

Identities:

  p(m*n,x) = p(m,x) + p(n,x),

  p(m*n,x) = p(gcd(m,n),x) + p(lcm(m,n),x),

  p(m+n,x) = p(gcd(m,n),x) + p((m+n)/gcd(m,n),x), so that if A003057 is read as a square matrix, then

  p(A003057,x) = p(A003989,x) + p(A106448,x).

Apart from powers of 3, all terms are even. - Charles R Greathouse IV, Feb 11 2012

Contains 2*p^m and p*2^m if p is an odd prime and m is in A052485. - Robert Israel, Oct 09 2016

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10566

EXAMPLE

Polynomials having nonnegative integer coefficients are matched to the positive integers as follows:

   n    p[n,x]    irreducible

  ---------------------------

   1    0         no

   2    1         no

   3    x         yes

   4    2         no

   5    x^2       no

   6    1+x       yes

   7    x^3       no

   8    3         no

   9    2x        yes

  10    1+x^2     yes

MAPLE

P:= n -> add(f[2]*x^(numtheory:-pi(f[1])-1), f =  ifactors(n)[2]):

select(irreduc @ P, [$1..200]); # Robert Israel, Oct 09 2016

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 *)

PROG

(PARI) is(n)=my(f=factor(n)); polisirreducible(sum(i=1, #f[, 1], f[i, 2]*'x^primepi(f[i, 1]-1))) \\ Charles R Greathouse IV, Feb 12 2012

CROSSREFS

Cf. A052485, A206285, A206296.

Positions of ones in A277322.

Terms of A277318 form a proper subset of this sequence. Cf. also A277316.

Sequence in context: A113502 A229307 A061904 * A268328 A247575 A289130

Adjacent sequences:  A206281 A206282 A206283 * A206285 A206286 A206287

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 05 2012

EXTENSIONS

Comment section edited and typos corrected by Antti Karttunen, Oct 09 2016

STATUS

approved

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Last modified July 18 21:25 EDT 2019. Contains 325144 sequences. (Running on oeis4.)