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Search: id:A160556
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| A160556 |
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Positive integers b for which the Diophantine equation f = (b^(2n) - b^n + 8n^2 - 2) / (2n * (2n + 1)) has at least ten solutions for n <= 10000 |
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+0
2
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| 2, 8, 14, 17, 26, 29, 32, 38, 41, 47, 50, 59, 62, 64, 65, 68, 74, 77, 83, 89, 95, 98, 101, 104, 110, 119, 122, 128, 131, 134, 137, 140, 143, 149, 152, 155, 161, 164, 167, 173, 179, 182, 185, 188, 194, 197, 200, 206, 209, 212, 215, 218, 221, 224, 227, 230, 233
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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For these equations (not exclusively), the sequences of 2n + 1 are dominated by primes.
When b = 2, there are 105 solutions with n less than 10000, and in this case, the sequence of n is also dominated by primes: only five of these are composite. The average difference between successive composite terms is near the magnitude of n. No composite values of 2n + 1 have been found. n and 2n + 1 account for roughly 3% of primes less than 20 billion. For other bases, n is almost always composite, and 2n + 1 is almost always prime.
The next most productive values of b less than 1000 are 509 (41 solutions) and 824 (40 solutions).
Bases that produce a greater or equal number of solutions than smaller bases, except 2, often have ones digit 4 or 9. Values of n associated with composite 2n + 1 are often divisible by 5.
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CROSSREFS
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Cf. A158034, A158035, A158036
Cf. A160557
Sequence in context: A002248 A050619 A056715 this_sequence A105610 A117104 A082933
Adjacent sequences: A160553 A160554 A160555 this_sequence A160557 A160558 A160559
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KEYWORD
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easy,nonn
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AUTHOR
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Reikku Kulon (reikku(AT)gmail.com), May 19 2009
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