A383748 a(n) = q is the smallest integer, such that the numbers -1/q, i/q, -i/q with i = sqrt(-1), are three zeros of the polynomial P(A783747(n),z) = Sum_{k=1..r} d(k)*z^(k-1) where d(1) < d(2), ..., < d(r) are the r divisors of A383747(n).

2, 3, 2, 2, 5, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 11, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 13, 2, 3, 2, 2, 2, 2, 3, 2, 2

Offset

1,1

Links

Table of n, a(n) for n=1..87.

Example

 n  q  m = A783747(n)         P(m,z)          3 zeros of P(m,z)
 1  2       8          1+2z+4z^2+8z^3         -1/2, -i/2, i/2
 2  3      27          1+3z+9z^2+27z^3        -1/3, -i/3, i/3
 3  2      88          1+2z+4z^2+8z^3+11z^4+  -1/2, -i/2, i/2
                       22z^5+44z^6+88z^7

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 n while (ii=0) do:
Q:=(x+1/q)*(x^2+1/q^2):
if divide(P, Q, 'R') then ii:=1:
A(n):=q:else fi:od:end proc:
seq(A(n), n=1..2500);

Crossrefs

Cf. A027750, A291127, A383747.

Sequence in context: A225176 A349271 A349387 * A118665 A333238 A336526

Adjacent sequences: A383745 A383746 A383747 * A383749 A383750 A383751

Keyword

nonn

Author

Michel Lagneau, May 08 2025

Status

approved