Michael De Vlieger, <a href="/A372323/b372323_1.txt">Table of n, a(n) for n = 3..10000</a>

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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.

allocated for Michael De VliegerA124652(n) is the a(n)-th term in row A372111(n-1) of irregular triangle A162306.

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

3,1

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

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

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

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

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

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

Analogous to A371910, which instead regards A109890 and A109735.

Michael De Vlieger, <a href="/A372323/b372323_1.txt">Table of n, a(n) for n = 3..10000</a>

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Michael De Vlieger, May 05 2024

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allocated for Michael De Vlieger

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