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

%I A292226 #27 Mar 14 2025 04:17:43
%S A292226 4,9,12,16,24,25,30,36,40,45,48,49,56,60,63,64,70,72,80,81,84,90,96,
%T A292226 105,108,112,120,121,126,132,135,140,144,150,154,160,165,168,169,175,
%U A292226 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
%N A292226 Composite numbers m (in increasing order) for which the m-th row polynomial of A027750 in rising powers is irreducible over the integers.
%C A292226 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.
%C A292226 P(1, x) = 1 (constant) and P(prime(n), x) = 1 + prime(n)*x are trivial.
%C A292226 The other polynomials corresponding to composite numbers from A002808 but not in the present sequence factorize into integer polynomials.
%C A292226 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, ...
%C A292226 From _Robert Israel_, Oct 31 2017: (Start)
%C A292226 Contains p^(q-1) if p is prime and q is an odd prime.
%C A292226 Disjoint from A006881. (End)
%H A292226 Robert Israel, <a href="/A292226/b292226.txt">Table of n, a(n) for n = 1..10000</a>
%e A292226 n = 1: P(4, x) = 1 + 2*x + 4*x^2 of degree A032741(4) = 2.
%e A292226 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).
%e A292226 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.
%p A292226 filter:= proc(n) local d,i,x;
%p A292226   if isprime(n) then return false fi;
%p A292226   d:= numtheory:-divisors(n);
%p A292226   irreduc(add(d[i]*x^(i-1),i=1..nops(d)))
%p A292226 end proc:
%p A292226 select(filter, [$2..1000]); # _Robert Israel_, Oct 31 2017
%t A292226 P[n_, x_] := (d = Divisors[n]).x^Range[0, Length[d] - 1];
%t A292226 okQ[n_] := CompositeQ[n] && IrreduciblePolynomialQ[P[n, x]];
%t A292226 Select[Range[300], okQ] (* _Jean-François Alcover_, Oct 30 2017 *)
%Y A292226 Cf. A006881, A027750, A032741, A291127.
%K A292226 nonn
%O A292226 1,1
%A A292226 _Wolfdieter Lang_, Oct 29 2017