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A372323 A124652(n) is the a(n)-th term in row A372111(n-1) of irregular triangle A162306.

%I A372323 #6 May 11 2024 21:59:00
%S A372323 2,4,4,4,5,7,5,8,8,2,10,8,12,11,13,6,13,6,6,9,8,11,4,8,16,5,6,7,13,12,
%T A372323 7,10,19,15,16,17,9,6,15,10,3,11,8,18,28,14,14,10,30,28,15,4,20,33,13,
%U A372323 12,6,22,18,21,12,11,29,12,11,8,24,18,8,14,17,32,33
%N A372323 A124652(n) is the a(n)-th term in row A372111(n-1) of irregular triangle A162306.
%C A372323 Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
%C A372323 Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.
%C A372323 Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.
%C A372323 Let T(k,j) represent the j-th term in row k of irregular triangle A162306.
%C A372323 a(n) = j is the position of b(n) in row s(n-1) of A162306.
%C A372323 b(n) = T(s(n-1), a(n)).
%C A372323 Analogous to A371910, which instead regards A109890 and A109735.
%H A372323 Michael De Vlieger, <a href="/A372323/b372323.txt">Table of n, a(n) for n = 3..10000</a>
%H A372323 Michael De Vlieger, <a href="/A372323/a372323.png">Bar chart showing a(n)/A372322(n-1)</a> for n = 3..1024. This chart illustrates the "depth" of A124652(n) among the terms of the A372111(n-1)-th row of A162306.
%e A372323 Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.
%e A372323 a(3) = 2 since b(3) = 3 is the 2nd term in row s(3) = 3 of A162306, {1, [3]}.
%e A372323 a(4) = 4 since b(4) = 4 is the 4th term in row s(4) = 6 of A162306, {1, 2, 3, [4], 6}.
%e A372323 a(5) = 4 since b(5) = 5 is T(s(n-1), 4) = T(10, 4), {1, 2, 4, [5], 8, 10}.
%e A372323 a(6) = 4 since b(6) = 9 is T(s(n-1), 4) = T(15, 4), {1, 3, 5, [9], 15}.
%e A372323 a(7) = 5 since b(7) = 6 is T(s(n-1), 5) = T(24, 5), {1, 2, 3, 4, [6], 8, 9, 12, 16, 18, 24}, etc.
%e A372323 Table relating this sequence to b = A124652, s = A372111, r = A372322, and A162306.
%e A372323    n b(n) s(n-1) a(n) r(n) row s(n-1) of A162306
%e A372323   ---------------------------------------------------------------------
%e A372323    3    3    3    2    2   {1, [3]}
%e A372323    4    4    6    4    5   {1, 2, 3, [4], 6}
%e A372323    5    5   10    4    6   {1, 2, 4, [5], 8, 10}
%e A372323    6    9   15    4    5   {1, 3, 5, [9], 15}
%e A372323    7    6   24    5   11   {1, 2, 3, 4, [6], ..., 24}
%e A372323    8    8   30    7   18   {1, 2, 3, 4, 5, 6, [8], ..., 30}
%e A372323    9   16   38    5    8   {1, 2, 4, 8, [16], 19, 32, 38}
%e A372323   10   12   54    8   16   {1, 2, 3, 4, 6, 8, 9, [12], ..., 54}
%e A372323   11   11   66    8   22   {1, 2, 3, 4, 6, 8, 9, [11], ..., 66}
%e A372323   12    7   77    2    5   {1, [7], 11, 49, 77}
%e A372323   13   14   84   10   28   {1, 2, 3, 4, ..., 12, [14], ..., 84}
%e A372323   14   28   98    8   13   {1, 2, 4, 7, ..., 16, [28], ..., 98}
%t A372323 nn = 75; c[_] := False;
%t A372323 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A372323 f[x_] := Select[Range[x], Divisible[x, rad[#]] &];
%t A372323 Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];
%t A372323 Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];
%t A372323   Sow[FirstPosition[r, k][[1]]]; c[k] = True;
%t A372323   s += k, {i, 3, nn}] ][[-1, 1]]
%Y A372323 Cf. A007947, A010846, A124652, A162306, A371910, A372111, A372322.
%K A372323 nonn
%O A372323 3,1
%A A372323 _Michael De Vlieger_, May 05 2024