%I #6 May 11 2024 21:59:00

%S 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 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 12,6,22,18,21,12,11,29,12,11,8,24,18,8,14,17,32,33

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

%C Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.

%C Let r(x) = A010846(x), the number of m <= x such that rad(m) | x, where rad = A007947.

%C Let row k of A162306 contain { m : rad(m) | k, m <= k }. Thus r(k) is the length of row k of A162306.

%C Let T(k,j) represent the j-th term in row k of irregular triangle A162306.

%C a(n) = j is the position of b(n) in row s(n-1) of A162306.

%C b(n) = T(s(n-1), a(n)).

%C Analogous to A371910, which instead regards A109890 and A109735.

%H Michael De Vlieger, <a href="/A372323/b372323.txt">Table of n, a(n) for n = 3..10000</a>

%H 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 Let b(x) = A124652(x) and let s(x) = A372111(x), where A372111 contains partial sums of A124652.

%e a(3) = 2 since b(3) = 3 is the 2nd term in row s(3) = 3 of A162306, {1, [3]}.

%e a(4) = 4 since b(4) = 4 is the 4th term in row s(4) = 6 of A162306, {1, 2, 3, [4], 6}.

%e a(5) = 4 since b(5) = 5 is T(s(n-1), 4) = T(10, 4), {1, 2, 4, [5], 8, 10}.

%e a(6) = 4 since b(6) = 9 is T(s(n-1), 4) = T(15, 4), {1, 3, 5, [9], 15}.

%e 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 Table relating this sequence to b = A124652, s = A372111, r = A372322, and A162306.

%e n b(n) s(n-1) a(n) r(n) row s(n-1) of A162306

%e ---------------------------------------------------------------------

%e 3 3 3 2 2 {1, [3]}

%e 4 4 6 4 5 {1, 2, 3, [4], 6}

%e 5 5 10 4 6 {1, 2, 4, [5], 8, 10}

%e 6 9 15 4 5 {1, 3, 5, [9], 15}

%e 7 6 24 5 11 {1, 2, 3, 4, [6], ..., 24}

%e 8 8 30 7 18 {1, 2, 3, 4, 5, 6, [8], ..., 30}

%e 9 16 38 5 8 {1, 2, 4, 8, [16], 19, 32, 38}

%e 10 12 54 8 16 {1, 2, 3, 4, 6, 8, 9, [12], ..., 54}

%e 11 11 66 8 22 {1, 2, 3, 4, 6, 8, 9, [11], ..., 66}

%e 12 7 77 2 5 {1, [7], 11, 49, 77}

%e 13 14 84 10 28 {1, 2, 3, 4, ..., 12, [14], ..., 84}

%e 14 28 98 8 13 {1, 2, 4, 7, ..., 16, [28], ..., 98}

%t nn = 75; c[_] := False;

%t rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];

%t f[x_] := Select[Range[x], Divisible[x, rad[#]] &];

%t Array[Set[{a[#], c[#]}, {#, True}] &, 2]; s = a[1] + a[2];

%t Reap[Do[r = f[s]; k = SelectFirst[r, ! c[#] &];

%t Sow[FirstPosition[r, k][[1]]]; c[k] = True;

%t s += k, {i, 3, nn}] ][[-1, 1]]

%Y Cf. A007947, A010846, A124652, A162306, A371910, A372111, A372322.

%K nonn

%O 3,1

%A _Michael De Vlieger_, May 05 2024