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 A085098 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = 2. 4
 1, 1, 5, 43, 876, 49513, 13005235 (list; graph; refs; listen; history; text; internal format)
 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,changed 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

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Last modified September 9 20:39 EDT 2024. Contains 375765 sequences. (Running on oeis4.)