A174057 Semi-sums (means) of a Fermat prime and a Mersenne prime.

3, 4, 5, 6, 10, 12, 17, 18, 24, 65, 66, 72, 130, 132, 144, 192, 4097, 4098, 4104, 4224, 32770, 32772, 32784, 32832, 36864, 65537, 65538, 65544, 65664, 98304, 262145, 262146, 262152, 262272, 294912, 1073741825, 1073741826, 1073741832

Offset

1,1

Comments

The subsequence of prime semi-sums (means) of a Fermat prime and a Mersenne prime begins: 3, 5, 17, 65537 = (3 + 131071)/2. R. J. Mathar, on the remaining primes in the half sum, searched through all sums that can be created from the existing values of the two OEIS sequences, and that the next Fermat prime is known to be > 2^(2^32) + 1. So it is safe to say that the next prime > 65537 in the half sum (if it exists) is larger than 85070591730234615865843651857942085632, because adding that huge next Fermat prime would lead to even larger numbers. Of course one could easily boost that estimate by using the b-file of A000668.

Links

Table of n, a(n) for n=1..38.

Formula

{(A019434(i) + A000668(j))/2}. {(((2^p)-1) + (2^(2^k)+1))/2 = 2^(p-1) + 2^((2^k)-1) for p in A000043 and k in {0,1,2,3,4}}.

Example

a(1) = 3 = half of first Mersenne prime + first Fermat prime = (3+3)/2.
a(2) = 4 = half of first Mersenne prime + 2nd Fermat prime = (3+5)/2.
a(3) = 5 = half of 2nd Mersenne prime + first Fermat prime = (7+3)/2.
a(4) = 6 = half of 2nd Mersenne prime + 2nd Fermat prime = (7+5)/2.
a(5) = 10 = half of 2nd Mersenne prime + 3rd Fermat prime = (3+17)/2.

Crossrefs

Cf. A000668, A171251-A171255, A155877 Sums of three Fermat numbers, A166484 Prime sums of three Fermat numbers, A174055, A174056.

Sequence in context: A263118 A219041 A218946 * A335231 A164977 A103033

Adjacent sequences: A174054 A174055 A174056 * A174058 A174059 A174060

Keyword

nonn

Author

Jonathan Vos Post, Mar 06 2010

Status

approved