%I #10 Mar 15 2024 11:59:01

%S 4,8,4,4,4,16,8,4,8,4,8,16,8,4,8,24,16,8,8,16,8,8,8,4,4,8,16,8,4,16,

%T 24,8,4,8,8,8,16,8,8,4,16,16,8,16,8,16,8,16,16,8,32,8,4,4,8,24,16,8,8,

%U 4,8,16,4,4,16,32,16,8,32,4,16,32,8,4,8,32,8,4,32,4,12,8,16,4,12,32,8,8,8

%N Number of integers m of the form (1+2x+4x^3)/(x+n).

%C The partial fraction decomposition (1+2x+4x^3)/(x+n)=4x^2-4nx+2+4n^2+(1-2n-4n^3)/(x+n) demonstrates that one can generate the solutions for any n by searching through all positive and negative divisors of 1-2n-4n^3, which are set to x+n, such that at least one solution (from the divisor 1, or -1, or 1-2n-4n^3, or -1+2n+4n^3, which may be degenerate) must exist. - _R. J. Mathar_, Oct 21 2006

%e Values of m=(1+2x+4x^3)/(x+n):

%e n, {m_i}

%e 1,{1,35,53,175},

%e 2,{-5,23,73,113,277,419,4109,5791},

%e 3,{-35,263,47117,55255},

%e 4,{-113,509,264245,289495},

%e 5,{-263,875,1006085,1067167},

%e 6,{-509,-1,1,1069,1099,1259,1385,2789,4769,7879,53927,71941,110329,135539,2999933,3125935},

%e 7,{-875,-7,1387,2063,284233,330779,7557149,7789831},

%e 8,{-1385,2933,16826597,17222695},

%e 9,{-2063,-5,2345,4019,657959,748477,34094165,34727695},

%e 10,{-2933,5345,64128365,65092927}.

%e Corresponding values of x's:

%e n,{x_i}

%e 1,{0,-2,4,-6},

%e 2,{-1,3,5,-3,-7,-9,33,-37},

%e 3,{-2,-4,110,-116},

%e 4,{-3,-5,259,-267},

%e 5,{-4,-6,504,-514},

%e 6,{-5,-1,1,-11,19,-13,-7,29,-31,-41,119,-131,169,-181,869,-881},

%e 7,{-6,-2,-12,-8,270,-284,1378,-1392},

%e 8,{-7,-9,2055,-2071},

%e 9,{-8,-2,-16,-10,410,-428,2924,-2942},

%e 10,{-9,-11,4009,-4029}.

%p A122110 := proc(n) local allm,dvs,i,x,m ; allm := {} : dvs := numtheory[divisors](1-2*n-4*n^3) : for i from 1 to nops(dvs) do x := op(i,dvs)-n ; m := (1+2*x+4*x^3)/(x+n) ; allm := allm union {m} ; x := -op(i,dvs)-n ; m := (1+2*x+4*x^3)/(x+n) ; allm := allm union {m} ; od ; RETURN(nops(allm)) ; end : for n from 1 to 100 do printf("%d,",A122110(n)) ; od ; # _R. J. Mathar_, Oct 21 2006

%K nonn

%O 1,1

%A _Zak Seidov_, Oct 18 2006

%E More terms from _R. J. Mathar_, Oct 21 2006