What are the next three numbers in this sequence? 6, 28, 496, 8128, ... ? Please explain how you determined these three numbers.

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