Integers of the form (1+2x+4x^3)/(x+n)

[Richard Mathar]

zs> From seqfan-owner at ext.jussieu.fr  Wed Oct 18 22:47:32 2006
zs> Return-Path: <seqfan-owner at ext.jussieu.fr>
zs> Date: Wed, 18 Oct 2006 12:47:02 -0700 (PDT)
zs> From: zak seidov <zakseidov at yahoo.com>
zs> Subject: Integers of the form  (1+2x+4x^3)/(x+n)
zs> To: seqfan at ext.jussieu.fr
zs> 
zs> Not submitted yet.
zs>  
zs> My Qs to gurus:
zs> 
zs> No solutions for n>10?
zs> Any theory behind all this?
zs> 
zs> Thanks, Zak
zs> 
zs> %I A000001
zs> %S A000001 4,8,4,4,4,16,8,4,8,4
zs> %N A000001 Number of integers m of the form
zs> (1+2x+4x^3)/(x+n)
zs> %C A000001 No solutions for n>10?
zs> %e A000001 
zs> 
zs> Values of m=(1+2x+4x^3)/(x+n):
zs> n, {m_i}
zs> 1,{1,35,53,175},
zs> 2,{-5,23,73,113,277,419,4109,5791},
zs> 3,{-35,263,47117,55255},
zs> 4,{-113,509,264245,289495},
zs> 5,{-263,875,1006085,1067167},
zs> 6,{-509,-1,1,1069,1099,1259,1385,2789,4769,7879,53927,71941,110329,135539,2999933,3125935},
zs> 7,{-875,-7,1387,2063,284233,330779,7557149,7789831},
zs> 8,{-1385,2933,16826597,17222695},
zs> 9,{-2063,-5,2345,4019,657959,748477,34094165,34727695},
zs> 10,{-2933,5345,64128365,65092927}.
zs> 
zs> Corresponding x's:
zs> n,{x_i}
zs> 1,{0,-2,4,-6},
zs> 2,{-1,3,5,-3,-7,-9,33,-37},
zs> 3,{-2,-4,110,-116},
zs> 4,{-3,-5,259,-267},
zs> 5,{-4,-6,504,-514},
zs> 6,{-5,-1,1,-11,19,-13,-7,29,-31,-41,119,-131,169,-181,869,-881},
zs> 7,{-6,-2,-12,-8,270,-284,1378,-1392},
zs> 8,{-7,-9,2055,-2071},
zs> 9,{-8,-2,-16,-10,410,-428,2924,-2942},
zs> 10,{-9,-11,4009,-4029}.
zs> 
zs> %O A000001 1
zs> %K A000001 ,nonn,
zs> %A A000001 Zak Seidov  (zakseidov at yahoo.com), Oct 18
zs> 2006

Rewrite
 m=(1+2x+4x^3)/(x+n):
by multiplication with x+n as
 m*x+m*n=1+2x+4x^3
then subtract m*x
 m*n=1+2x+4x^3-m*x
then divide through m
 n=(1+2x+4x^3)/m-x
Solutions with huge n are found by insertion of x=1,2,3,4,5,6,7,etc,
factorization of 1+2x+4x^3 (eg. x=14, 1+2x+4x^3=11005=5*31*71)
and dividing through a not too large of these factors representing m,
for example m=31 with x=14.
In this example x=14, m=31, n=11005/31-14=341 
or (1+x+4x^3)/(x+n)=11005/(14+341)=11005/355=31.
Other examples of these factorizations are 
x=19: 1+2x+4x^3=5^2*7*157 and one may chose m=5 or 25 or 7 or...
So I'd generally wrap the theory around the factorization of 1+2x+4x^3.

Hope I'm not too tired and this is at least partially correct...
RJM