A292226 Composite numbers m (in increasing order) for which the m-th row polynomial of A027750 in rising powers is irreducible over the integers.

4, 9, 12, 16, 24, 25, 30, 36, 40, 45, 48, 49, 56, 60, 63, 64, 70, 72, 80, 81, 84, 90, 96, 105, 108, 112, 120, 121, 126, 132, 135, 140, 144, 150, 154, 160, 165, 168, 169, 175, 176, 180, 182, 189, 192, 195, 198, 200, 208, 210, 216, 220, 224, 225, 231, 234, 240, 252, 260, 264, 270, 273, 275, 280, 286, 288, 289, 297, 300

Offset: 1

Author

Wolfdieter Lang, Oct 29 2017

Keywords

• nonn

Comments

The considered integer polynomials of degree A032741(a(n)) are P(a(n), x) = Sum_{k=0..A032741(a(n))} A027750(a(n), k+1)*x^k for n >= 1.

P(1, x) = 1 (constant) and P(prime(n), x) = 1 + prime(n)*x are trivial.

The other polynomials corresponding to composite numbers from A002808 but not in the present sequence factorize into integer polynomials.

This entry was motivated by the proposal A291127 by Michel Lagneau giving the numbers m for which P(m, x) = Sum_{k=0..A032741(m)} A027750(m, k+1)*x^k has at least two purely imaginary zeros. The present composite a(n) numbers do not appear in A291127. Other composite numbers also do not appear, like 18, 20, 28, 32, 44, ...

From Robert Israel, Oct 31 2017: (Start)

Contains p^(q-1) if p is prime and q is an odd prime.

Disjoint from A006881. (End)

Examples

			n = 1: P(4, x) = 1 + 2*x + 4*x^2 of degree A032741(4) = 2.
The composite number 6 is not a member of this sequence because P(6, x) = 1 + 2*x + 3*x^2 + 6*x^3 of degree A032741(6) = 3 factorizes as (1 + 2*x)*(1 + 3*x^2).
m = 18 is not a member of the sequence because P(18, x) = 1 + 2*x + 3*x^2 + 6*x^3 + 9*x^4 + 18*x^5 = (1 + 2*x)*(1 + 3*x^2 + 9*x^4). m = 18 does also not appear in A291127.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006881, A027750, A032741, A291127.

Programs

Maple

filter:= proc(n) local d,i,x;
  if isprime(n) then return false fi;
  d:= numtheory:-divisors(n);
  irreduc(add(d[i]*x^(i-1),i=1..nops(d)))
end proc:
select(filter, [$2..1000]); # Robert Israel, Oct 31 2017

Mathematica

P[n_, x_] := (d = Divisors[n]).x^Range[0, Length[d] - 1];
okQ[n_] := CompositeQ[n] && IrreduciblePolynomialQ[P[n, x]];
Select[Range[300], okQ] (* Jean-François Alcover, Oct 30 2017 *)