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COMMENTS
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The sequence is infinite because the powers of 2 are in the sequence.
Conjecture: the corresponding numbers k of the sequence is a sequence b(n) of powers of 2.
The sequence b(n) begins with 1, 1, 2, 2, 2, 4, 4, 8, 8, 8, 8, 8, 16, 16, 16, 16, 32, 32, 32, 32, 64, 64, 64, 64, 128, 128, 128, 128, 128, 128, 128, 128, 128, 256, 256, 256, 256, 256, 256, 256, 256, 512, 512, 512, 512, 512, 512, 512, 512, 1024, 1024, 1024, 1024, ...
The number of terms of the form 2^i is given by the sequence c(i,2^i) = 2, 3, 2, 5, 4, 4, 4, 9, 8, 8, 8, 8, 8, 8, 8, 17, ... for i = 0, 1, 2, ...
Property of the prime factors of a(n) for n > 1:
We observe periodic sequences of prime factors of the form 2^m + 1.
+---------------------------------------+------------------------------+
| subsequences of consecutive | corresponding prime factors |
| values of a(n) | |
+---------------------------------------+------------------------------+
|4, 6 |{2},{2, 3} |
|8, 12 |{2},{2, 3} |
|16, 20, 24, 30 |{2}, {2, 5}, {2, 3}, {2, 3, 5}|
|32, 40, 48, 60 |{2}, {2, 5}, {2, 3}, {2, 3, 5}|
|64, 80, 96, 120 |{2}, {2, 5}, {2, 3}, {2, 3, 5}|
|128, 160, 192, 240 |{2}, {2, 5}, {2, 3}, {2, 3, 5}|
|256, 272, 320, 340, 384, 408, 480, 510 |{2}, {2, 17},...,{2, 3, 5, 17}|
|512, 544, 640, 680, 768, 816, 960, 1020|{2}, {2, 17},...,{2, 3, 5, 17}|
|1024, 1088, 1280, 1360, 1536, ..., 2040|{2}, {2, 17},...,{2, 3, 5, 17}|
|2048, 2176, 2560, 2720, 3072, ..., 4080|{2}, {2, 17},...,{2, 3, 5, 17}|
|4096, 4352, 5120, 5440, 6144, ..., 8160|{2}, {2, 17},...,{2, 3, 5, 17}|
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