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).

8, 27, 88, 104, 125, 128, 136, 152, 184, 232, 248, 296, 328, 343, 344, 376, 424, 472, 488, 536, 568, 584, 632, 664, 712, 776, 783, 808, 824, 837, 856, 872, 904, 968, 999, 1016, 1048, 1096, 1107, 1112, 1161, 1192, 1208, 1256, 1269, 1304, 1331, 1336, 1352, 1384, 1431

Offset

1,1

Comments

Subsequence of A291127.

The corresponding integers q are in A383748.

Links

Table of n, a(n) for n=1..51.

Example

  n   m  P(m,z)                                  3 zeros of P(m,z)
  1   8  1+2z+4z^2+8z^3                          -1/2, -i/2, i/2
  2  27  1+3z+9z^2+27z^3                         -1/3, -i/3, i/3
  3  88  1+2z+4z^2+8z^3+11z^4+22z^5+44z^6+88z^7  -1/2, -i/2, i/2

Maple

with(numtheory) :
A:=proc(n) local P, Q, i, q, d, ii:
d:=divisors(n):P:=add(op(i, d)*x^(i-1), i=1..nops(d)):
ii:=0:for q from 1 to 10^4 while (ii=0) do:
Q:=(x+1/q)*(x^2+1/q^2):
if divide(P, Q, 'R') then ii:=1:
A(n):=n:else fi:od:end proc:
seq(A(n), n=1..2500);

Crossrefs

Cf. A027750, A291127, A383748.

Sequence in context: A212745 A229423 A166867 * A343323 A079712 A389108

Adjacent sequences: A383744 A383745 A383746 * A383748 A383749 A383750

Keyword

nonn

Author

Michel Lagneau, May 08 2025

Status

approved