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Flawless Series? (Posted on 2003-11-29) Difficulty: 3 of 5
What are the next three numbers in this sequence? 6, 28, 496, 8128, ... ? Please explain how you determined these three numbers.

See The Solution Submitted by SilverKnight    
Rating: 2.8889 (9 votes)

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Solution In fact... (another solution) | Comment 13 of 67 |
The relation to the Mersenne primes brings to mind that the first four terms of this sequence do not fully identify it. The sequence does in fact show 2^(p-1)*(2^p-1) for the first four primes. It just happens that for these four primes, 2^p-1 is also prime. However, if we choose to continue the series without regard to whether 2^p-1 is prime, we get a different continuation:

2 6
3 28
5 496
7 8128
11 2096128
13 33550336
17 8589869056
19 137438691328
23 35184367894528
29 144115187807420416
OK

as produced by the following program:
list
10 P=1
20 for J=1 to 10
30    P=nxtprm(P)
40    print P,2^(P-1)*(2^P-1)
50 next

so the next three numbers would be 2096128, 33550336, 8589869056, the first of which is imperfect as it does not involve a Mersenne prime.

  Posted by Charlie on 2003-11-30 11:16:54
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