

A122110


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


1



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, 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, 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
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OFFSET

1,1


COMMENTS

The partial fraction decomposition (1+2x+4x^3)/(x+n)=4x^24nx+2+4n^2+(12n4n^3)/(x+n) demonstrates that one can generate the solutions for any n by searching through all positive and negative divisors of 12n4n^3, which are set to x+n, such that at least on solution (from the divisor 1, or 1, or 12n4n^3, or 1+2n+4n^3, which may be degenerate) must exist.  R. J. Mathar, Oct 21 2006


LINKS

Table of n, a(n) for n=1..89.


EXAMPLE

Values of m=(1+2x+4x^3)/(x+n):
n, {m_i}
1,{1,35,53,175},
2,{5,23,73,113,277,419,4109,5791},
3,{35,263,47117,55255},
4,{113,509,264245,289495},
5,{263,875,1006085,1067167},
6,{509,1,1,1069,1099,1259,1385,2789,4769,7879,53927,71941,110329,135539,2999933,3125935},
7,{875,7,1387,2063,284233,330779,7557149,7789831},
8,{1385,2933,16826597,17222695},
9,{2063,5,2345,4019,657959,748477,34094165,34727695},
10,{2933,5345,64128365,65092927}.
Corresponding values of x's:
n,{x_i}
1,{0,2,4,6},
2,{1,3,5,3,7,9,33,37},
3,{2,4,110,116},
4,{3,5,259,267},
5,{4,6,504,514},
6,{5,1,1,11,19,13,7,29,31,41,119,131,169,181,869,881},
7,{6,2,12,8,270,284,1378,1392},
8,{7,9,2055,2071},
9,{8,2,16,10,410,428,2924,2942},
10,{9,11,4009,4029}.


MAPLE

A122110 := proc(n) local allm, dvs, i, x, m ; allm := {} : dvs := numtheory[divisors](12*n4*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


CROSSREFS

Sequence in context: A092511 A045816 A085991 * A082632 A296481 A155874
Adjacent sequences: A122107 A122108 A122109 * A122111 A122112 A122113


KEYWORD

nonn


AUTHOR

Zak Seidov, Oct 18 2006


EXTENSIONS

More terms from R. J. Mathar, Oct 21 2006


STATUS

approved



