This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A085098 #13 Sep 05 2024 21:34:32
%S A085098 1,1,5,43,876,49513,13005235
%N A085098 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = 2.
%C A085098 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
%e A085098 For n = 1, a(1) = 1, one solution: {x_1} = {1}.
%e A085098 For n = 2, a(2) = 1, one solution: {x_1, x_2} = {2, 3}.
%e A085098 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}.
%e A085098 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.
%p A085098 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:
%Y A085098 Cf. A118086.
%K A085098 hard,more,nonn
%O A085098 1,3
%A A085098 _Philippe Deléham_, Aug 10 2003
%E A085098 a(5) corrected and a(6) computed by _Gene Ward Smith_, Apr 11 2006
%E A085098 a(7) was found by _Hugo van der Sanden_, Dec 14 2007
%E A085098 Edited by _N. J. A. Sloane_, Oct 18 2008 at the suggestion of _Max Alekseyev_