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%I #24 Nov 15 2017 05:00:11
%S 4,9,12,16,24,25,30,36,40,45,48,49,56,60,63,64,70,72,80,81,84,90,96,
%T 105,108,112,120,121,126,132,135,140,144,150,154,160,165,168,169,175,
%U 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 Composite numbers m (in increasing order) for which the m-th row polynomial of A027750 in rising powers is irreducible over the integers.
%C 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 P(1, x) = 1 (constant) and P(prime(n)), x) = 1 + prime(n)*x are trivial.
%C The other polynomials corresponding to composite numbers from A002808 but not in the present sequence factorize into integer polynomials.
%C 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 From _Robert Israel_, Oct 31 2017: (Start)
%C Contains p^(q-1) if p is prime and q is an odd prime.
%C Disjoint from A006881. (End)
%H Robert Israel, <a href="/A292226/b292226.txt">Table of n, a(n) for n = 1..10000</a>
%e n = 1: P(4, x) = 1 + 2*x + 4*x^2 of degree A032741(4) = 2.
%e 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 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 filter:= proc(n) local d,i,x;
%p if isprime(n) then return false fi;
%p d:= numtheory:-divisors(n);
%p irreduc(add(d[i]*x^(i-1),i=1..nops(d)))
%p end proc:
%p select(filter, [$2..1000]); # _Robert Israel_, Oct 31 2017
%t P[n_, x_] := (d = Divisors[n]).x^Range[0, Length[d] - 1];
%t okQ[n_] := CompositeQ[n] && IrreduciblePolynomialQ[P[n, x]];
%t Select[Range[300], okQ] (* _Jean-François Alcover_, Oct 30 2017 *)
%Y Cf. A006881, A027750, A032741, A291127.
%K nonn
%O 1,1
%A _Wolfdieter Lang_, Oct 29 2017
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