cp's OEIS Frontend

A383747 Consider the polynomial P(m,z) = Sum_{k=1..r} d(k)*z^(k-1) where d(1) < d(2) < ... < d(r) are the r divisors of m. The sequence lists the numbers m such that P(m,z) contains at least three zeros of the form -1/q, i/q, -i/q, for some integer q, i = sqrt(-1).

%I A383747 #11 May 16 2025 18:53:05
%S A383747 8,27,88,104,125,128,136,152,184,232,248,296,328,343,344,376,424,472,
%T A383747 488,536,568,584,632,664,712,776,783,808,824,837,856,872,904,968,999,
%U A383747 1016,1048,1096,1107,1112,1161,1192,1208,1256,1269,1304,1331,1336,1352,1384,1431
%N A383747 Consider the polynomial P(m,z) = Sum_{k=1..r} d(k)*z^(k-1) where d(1) < d(2) < ... < d(r) are the r divisors of m. The sequence lists the numbers m such that P(m,z) contains at least three zeros of the form -1/q, i/q, -i/q, for some integer q, i = sqrt(-1).
%C A383747 Subsequence of A291127.
%C A383747 The corresponding integers q are in A383748.
%e A383747   n   m  P(m,z)                                  3 zeros of P(m,z)
%e A383747   1   8  1+2z+4z^2+8z^3                          -1/2, -i/2, i/2
%e A383747   2  27  1+3z+9z^2+27z^3                         -1/3, -i/3, i/3
%e A383747   3  88  1+2z+4z^2+8z^3+11z^4+22z^5+44z^6+88z^7  -1/2, -i/2, i/2
%p A383747 with(numtheory) :
%p A383747 A:=proc(n) local P, Q, i, q, d, ii:
%p A383747 d:=divisors(n):P:=add(op(i,d)*x^(i-1),i=1..nops(d)):
%p A383747 ii:=0:for q from 1 to 10^4 while (ii=0) do:
%p A383747 Q:=(x+1/q)*(x^2+1/q^2):
%p A383747 if divide(P,Q,'R') then ii:=1:
%p A383747 A(n):=n:else fi:od:end proc:
%p A383747 seq(A(n), n=1..2500);
%Y A383747 Cf. A027750, A291127, A383748.
%K A383747 nonn
%O A383747 1,1
%A A383747 _Michel Lagneau_, May 08 2025