cp's OEIS Frontend

A381097 Consider the polynomial P(m,z) = Sum_{i=1..k} d(i)*z^(i-1) where d(1), d(2), ..., d(k) are the k divisors of m. The sequence lists the numbers m such that P(m,z) is irreducible.

%I A381097 #19 Feb 26 2025 09:04:18
%S A381097 2,3,4,5,7,9,11,12,13,16,17,19,23,24,25,29,30,31,36,37,40,41,43,45,47,
%T A381097 48,49,53,56,59,60,61,63,64,67,70,71,72,73,79,80,81,83,84,89,90,96,97,
%U A381097 101,103,105,107,108,109,112,113,120,121,126,127,131,132,135
%N A381097 Consider the polynomial P(m,z) = Sum_{i=1..k} d(i)*z^(i-1) where d(1), d(2), ..., d(k) are the k divisors of m. The sequence lists the numbers m such that P(m,z) is irreducible.
%C A381097 The squares>1 and the prime numbers are in the sequence.
%e A381097 The prime numbers q are in the sequence because P(q,z) = qz + 1.
%e A381097 6 is not in the sequence because P(6,z)=(2z+1)*(3z^2+1).
%e A381097 The following table gives the irreducible polynomials.
%e A381097 +-----------------------------------------------------------+
%e A381097 |  m |                 P(m,z)                               |
%e A381097 +-----------------------------------------------------------+
%e A381097 |  4 | 1 + 2z + 4z^2                                        |
%e A381097 +-----------------------------------------------------------+
%e A381097 |  9 | 1 + 3z + 9z^2                                        |
%e A381097 +-----------------------------------------------------------+
%e A381097 | 12 | 1 + 2z + 3z^2 + 4z^3 + 6z^4 + 12z^5                  |
%e A381097 +-----------------------------------------------------------+
%e A381097 | 16 | 1 + 2z + 4z^2 + 8z^3 + 16z^4                         |
%e A381097 +----------------------------+------------------------------+
%e A381097 | 24 | 1 + 2z + 3z^2 + 4z^3 + 6z^4 + 8z^5 + 12z^6 + 24z^7   |
%e A381097 +-----------------------------------------------------------+
%e A381097 | 25 | 1 + 5z + 25z^2                                       |
%e A381097 +-----------------------------------------------------------+
%e A381097 | 30 | 1 + 2z + 3z^2 + 5z^3 + 6z^4 + 10z^5 + 15z^6 + 30z^7  |
%e A381097 +-----------------------------------------------------------+
%p A381097 with(numtheory):
%p A381097 for n from 1 to 135 do :
%p A381097   it:=0:d:=divisors(n):P:=add(op(i,d)*x^(i-1),i=1..nops(d)):
%p A381097    y:=fsolve(P,x,complex):z:=evalf({%}):k:=nops(z):
%p A381097     if irreduc(P) then printf(`%d, `,n):else fi:
%p A381097 od:
%o A381097 (PARI) isok(n) = my(d=divisors(n)); polisirreducible(sum(i=1, #d, d[i]*z^(i-1))); \\ _Michel Marcus_, Feb 14 2025
%Y A381097 Cf. A291127.
%K A381097 nonn
%O A381097 1,1
%A A381097 _Michel Lagneau_, Feb 14 2025