

A085098


a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = 2.


4




OFFSET

1,3


COMMENTS

Number of ways 2 is a product of n superparticular ratios, without regard to order. A superparticular ratio is a ratio of the form m/(m1). The question relates to music theory, in that various permutations of these products result in scales.  Gene Ward Smith, Apr 11 2006


LINKS



EXAMPLE

For n = 1, a(1) = 1, one solution: {x_1} = {1}.
For n = 2, a(2) = 1, one solution: {x_1, x_2} = {2, 3}.
For n = 3, a(3) = 5, five solutions: {x_1, x_2, x_3} = {3, 4, 5}, {3, 3, 8}, {2, 4, 15}, {2, 5, 9}, {2, 6, 7}.
In other words, a(3) = 5 since 2 can be written as (4/3)^2 (9/8), (4/3)(5/4)(6/5), (3/2)(7/6)(8/7), (3/2)(6/5)(10/9) or (3/2)(5/4)(16/15) but in no other way using superparticular rations.


MAPLE

spsubdiv := proc(r::rational, n::integer) # Maple program by David Canright local i, j, l, s; # option remember; if n=1 then if numer(r)=denom(r)+1 then [r] else ( NULL ) end if; else s := NULL; for i from floor(1/(r1))+1 while (1+1/i)^n >= r do l := [spsubdiv( r/(1+1/i), n1 )]; for j to nops(l) do if op(1, op(j, l)) <= (1+1/i) then s := s, [(1+1/i), op(op(j, l))]; end if od; od; s; end if; end: spl := proc(r, n) [spsubdiv(r, n)] end: spcount := proc(r, n) nops(spl(r, n)) end:


CROSSREFS



KEYWORD

hard,more,nonn,changed


AUTHOR



EXTENSIONS



STATUS

approved



