A085098 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = 2.

1, 1, 5, 43, 876, 49513, 13005235

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/(m-1). The question relates to music theory, in that various permutations of these products result in scales. - Gene Ward Smith, Apr 11 2006

Links

Table of n, a(n) for n=1..7.

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/(r-1))+1 while (1+1/i)^n >= r do l := [spsubdiv( r/(1+1/i), n-1 )]; 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

Cf. A118086.

Sequence in context: A255895 A160450 A114604 * A271679 A350875 A099794

Adjacent sequences: A085095 A085096 A085097 * A085099 A085100 A085101

Keyword

hard,more,nonn

Author

Philippe Deléham, Aug 10 2003

Extensions

a(5) corrected and a(6) computed by Gene Ward Smith, Apr 11 2006

a(7) was found by Hugo van der Sanden, Dec 14 2007

Edited by N. J. A. Sloane, Oct 18 2008 at the suggestion of Max Alekseyev

Status

approved