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A299240
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Ranks of {2,3}-power towers in which #2's > #3's; see Comments.
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3
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1, 3, 6, 8, 9, 10, 13, 14, 17, 19, 21, 27, 28, 29, 30, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 51, 55, 56, 57, 58, 59, 60, 61, 63, 71, 72, 73, 75, 79, 80, 81, 83, 87, 88, 89, 91, 95, 103, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124
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OFFSET
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1,2
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COMMENTS
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Suppose that S is a set of real numbers. An S-power-tower, t, is a number t = x(1)^x(2)^...^x(k), where k >= 1 and x(i) is in S for i = 1..k. We represent t by (x(1),x(2),...,x(k), which for k > 1 is defined as (x(1),((x(2),...,x(k-1)); (2,3,2) means 2^9. The number k is the *height* of t. If every element of S exceeds 1 and all the power towers are ranked in increasing order, the position of each in the resulting sequence is its *rank*. See A299229 for a guide to related sequences.
This sequence together with A299241 and A299242 partition the positive integers.
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LINKS
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EXAMPLE
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The first six terms are the ranks of these towers: t(1) = (2), t(3) = (2,2), t(6) = (2,2,2), t(8) = (3,2,2), t(9) = (2,2,3), t(10) = (2,3,2).
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MATHEMATICA
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t[1] = {2}; t[2] = {3}; t[3] = {2, 2}; t[4] = {2, 3}; t[5] = {3, 2};
t[6] = {2, 2, 2}; t[7] = {3, 3}; t[8] = {3, 2, 2}; t[9] = {2, 2, 3};
t[10] = {2, 3, 2}; t[11] = {3, 2, 3}; t[12] = {3, 3, 2};
z = 190; g[k_] := If[EvenQ[k], {2}, {3}]; f = 6;
While[f < 13, n = f; While[n < z, p = 1;
While[p < 12, m = 2 n + 1; v = t[n]; k = 0;
While[k < 2^p, t[m + k] = Join[g[k], t[n + Floor[k/2]]]; k = k + 1];
p = p + 1; n = m]]; f = f + 1]
Select[Range[1000], Count[t[#], 2] > Count[t[#], 3] &]; (* A299240 *)
Select[Range[1000], Count[t[#], 2] == Count[t[#], 3] &]; (* A299241 *)
Select[Range[1000], Count[t[#], 2] < Count[t[#], 3] &]; (* A299242 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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