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A355406
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Positive integers that are not powers of 2 and whose Collatz trajectory has maximum power of 2 different from 2^4.
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1
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21, 42, 75, 84, 85, 113, 150, 151, 168, 170, 201, 226, 227, 267, 300, 301, 302, 336, 340, 341, 401, 402, 403, 423, 452, 453, 454, 475, 534, 535, 537, 600, 602, 604, 605, 633, 635, 672, 680, 682, 713, 715, 802, 803, 804, 805, 806, 846, 847, 891, 904, 906, 908, 909, 950, 951, 953, 955
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OFFSET
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1,1
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COMMENTS
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It is conjectured that 15/16 (93.75%) of the positive integers that are not powers of 2 have 2^4 as the maximum power of 2 in their Collatz trajectory (see A232503 and A355187). {a(n)} lists the remaining positive integers. Consequently, it is conjectured that this sequence will have lim_{n->oo} a(n)/n = 1/16.
Among the numbers from 1 to 1000, there are 10 that are powers of 2, and there are 932 others (excluding 16) whose Collatz trajectories contain 2^4 as their maximum power of 2. The remaining 58 numbers are the first 58 terms of {a(n)}.
If k is in this sequence then so is k*2^j for any j > 0. To find a "primitive" set simply eliminate the even terms (see A350160).
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LINKS
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EXAMPLE
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21 is a term since its trajectory 21 64 32 16 8 2 1 has 64 as the highest power of 2, which is more than 16 and 21 is not itself a power of 2.
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MATHEMATICA
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collatz[n_] := Module[{}, If[OddQ[n], 3n+1, n/2]]; step[n_] := Module[{p=0, m=n, q}, While[!IntegerQ[q=Log[2, m]], m=collatz[m]; p++]; {p, q, n}]; Last/@Select[Table[step[n], {n, 1, 10^5}], #[[1]]>0 && #[[2]]!=4 &]
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CROSSREFS
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Subset of A308149 where terms that are powers of 2 have been omitted.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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