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A306012 Let S(m) = d(k)/d(1) + ... + d(1)/d(k), where d(1)..d(k) are the unitary divisors of m; then a(n) is the denominator of S(m) when all the numbers S(m) are arranged in increasing order.

%I A306012 #11 Feb 06 2019 11:42:54
%S A306012 1,2,3,4,5,7,8,3,9,11,1,13,6,16,17,3,7,19,10,9,23,21,25,27,12,11,29,
%T A306012 14,31,32,13,33,37,7,18,41,4,17,43,3,39,47,22,45,19,49,53,24,26,51,23,
%U A306012 55,28,59,21,61,5,57,64,63,67,27,1,71,2,29,73,3,36,69
%N A306012 Let S(m) = d(k)/d(1) + ... + d(1)/d(k), where d(1)..d(k) are the unitary divisors of m; then a(n) is the denominator of S(m) when all the numbers S(m) are arranged in increasing order.
%e A306012 The first 8 pairs {m,S(m)} are {1, 1}, {2, 5/2}, {3, 10/3}, {4, 17/4}, {5, 26/5}, {6, 25/3}, {7, 50/7}, {8, 65/8}. When the numbers S(m) are arranged in increasing order, the pairs are {1, 1}, {2, 5/2}, {3, 10/3}, {4, 17/4}, {5, 26/5}, {7, 50/7}, {8, 65/8}, {6, 25/3}, so that the first 8 denominators are 1,2,3,4,5,7,8,3.
%t A306012 z = 100; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &];
%t A306012 k[n_] := Length[r[n]];
%t A306012 t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];
%t A306012 s = Table[{n, Total[t[n]]}, {n, 1, z}]
%t A306012 v = SortBy[s, Last]
%t A306012 v1 = Table[v[[n]][[1]], {n, 1, z}]  (* A306010 *)
%t A306012 w = Table[v[[n]][[2]], {n, 1, z}];
%t A306012 Numerator[w]    (* A306011 *)
%t A306012 Denominator[w]  (* A306012 *)
%Y A306012 Cf. A077610, A229994, A229996, A305995, A306010, A306011.
%K A306012 nonn
%O A306012 1,2
%A A306012 _Clark Kimberling_, Jun 16 2018