(S459b)
  Subj:     Find the next number in the
          sequence 6, 28, 496, . . .

Three possible solutions are
8126, 130816, 8128, and 2849

 
 

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Solution One comes from my students

When I taught school my students decided the
best answer was that these were the first
three perfect numbers.  The next perfect
number is 8126.  A perfect number is the
sum of all it's factors.
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Solution Two comes from Anon Jr. and Jack

6, 28, 496, 130816 - This is 2**N - 2**M,
where M doubles every term and N2 = N1 + M2.
So

(2**3 - 2**1), (2**5 - 2**2), (2**9 - 2**4), (2**17 - 2**8) or

(  8   -    2 ), (  32  -   4  ), ( 512 -  16 ), (131072 - 256)

= 6, 28, 496, 130816
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Solution Three comes from Abe

For the non-wimpy sequence... 3*2=6, 7*4=28, 31*16=496, 127*64=8128

which is
Mersenne prime * largest power of 2 that is less than the Mersenne prime
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Solution Four comes from Jack

6, 28, 496, 2849 - the digits 6,2,8,4,9
repeat in order with one additional digit
in each subsequent number