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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 numerator of S(m) when all the numbers S(m) are arranged in increasing order.
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%I #12 Jun 25 2024 01:30:00

%S 1,5,10,17,26,50,65,25,82,122,13,170,85,257,290,52,125,362,221,205,

%T 530,500,626,730,325,305,842,425,962,1025,425,1220,1370,260,697,1682,

%U 169,725,1850,130,1700,2210,1037,2132,905,2402,2810,1285,1445,2900,1325

%N 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 numerator of S(m) when all the numbers S(m) are arranged in increasing order.

%e 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 numerators are 1,5,10,17,26,50,65,25.

%t z = 100; r[n_] := Select[Divisors[n], GCD[#, n/#] == 1 &];

%t k[n_] := Length[r[n]];

%t t[n_] := Table[r[n][[k[n] + 1 - i]]/r[n][[k[1] + i - 1]], {i, 1, k[n]}];

%t s = Table[{n, Total[t[n]]}, {n, 1, z}]

%t v = SortBy[s, Last]

%t v1 = Table[v[[n]][[1]], {n, 1, z}] (* A306010 *)

%t w = Table[v[[n]][[2]], {n, 1, z}];

%t Numerator[w] (* A306011 *)

%t Denominator[w] (* A306012 *)

%Y Cf. A077610, A229994, A229996, A305995, A306010, A306012.

%K nonn

%O 1,2

%A _Clark Kimberling_, Jun 16 2018