# [seqfan] Re: Sums of three Mersenne primes, and prime sums of three Mersenne primes

Maximilian Hasler maximilian.hasler at gmail.com
Fri Mar 5 09:34:57 CET 2010

```I think the fact that the set of sums of 2 Mersenne primes consists
only in even numbers does not justify that it should not be considered
(even "prior" to the sum of three).
Additionally, one could take one-half of these (i.e. averages between
two Mersenne primes)
which seems not in OEIS, either.

Sums of two (not necessarily distinct) Mersenne primes (A668)
S1 = [ 6, 10, 14, 34, 38, 62, 130, 134, 158, 254, 8194, 8198, 8222,
8318, 16382, 131074, 131078, 131102, 131198, 139262, 262142, 524290,
524294, 524318, 524414, 532478, 655358, 1048574, 2147483650,
2147483654, 2147483678, 2147483774, 2147491838, 2147614718,
2148007934, 4294967294, 2305843009213693954, 2305843009213693958,
2305843009213693982, 2305843009213694078, 2305843009213702142,
2305843009213825022, 2305843009214218238, 2305843011361177598,
4611686018427387902, 618970019642690137449562114,
618970019642690137449562118,... ]

Half-sum (= average) of two (not necessarily distinct) Mersenne primes (A668)
S2 = [ 3, 5, 7, 17, 19, 31, 65, 67, 79, 127, 4097, 4099, 4111, 4159,
8191, 65537, 65539, 65551, 65599, 69631, 131071, 262145, 262147,
262159, 262207, 266239, 327679, 524287, 1073741825, 1073741827,
1073741839, 1073741887, 1073745919, 1073807359, 1074003967,
2147483647, 1152921504606846977, 1152921504606846979,
1152921504606846991, 1152921504606847039, 1152921504606851071,
1152921504606912511, 1152921504607109119, 1152921505680588799,
2305843009213693951, 309485009821345068724781057,
309485009821345068724781059, ...]

%C A668 is a subsequence of the above S2
which motivates:

Half-sum (= average) of two distinct Mersenne primes (A668)
S3 = [5, 17, 19, 65, 67, 79, 4097, 4099, 4111, 4159, 65537, 65539,
65551, 65599, 69631, 262145, 262147, 262159, 262207, 266239, 327679,
1073741825, 1073741827, 1073741839, 1073741887, 1073745919,
1073807359, 1074003967, 1152921504606846977, 1152921504606846979,
1152921504606846991, 1152921504606847039, 1152921504606851071,
1152921504606912511, 1152921504607109119, 1152921505680588799,
309485009821345068724781057, 309485009821345068724781059,
309485009821345068724781071, 309485009821345068724781119, ;;;]

%C this is S2 \ A668.

then we also have
S4 = primes in S2 (again includes A668)
S5 = primes in S3 (starts 5,17,19,67,79,... which still is not in OEIS)

Maximilian

On Fri, Mar 5, 2010 at 8:25 AM, Jonathan Post <jvospost3 at gmail.com> wrote:
> Sums of five Mersenne primes can also be prime (though, obviously sums
> of an even number of Mersenne primes are even).
>
> 3 + 3 + 3 + 3 + 7 = 19
> 3 + 3 + 3 + 7 + 7 = 23
> 3 + 7 + 7 + 7 + 7 = 31
> 3 + 3 + 3 + 3 + 31 = 43
> 3 + 3 + 3 + 7 + 31 = 47
> 7 + 7 + 7 + 7 + 31 = 59
> 3 + 3 + 3 + 31 + 31 = 71
> 3 + 7 + 7+ 31 + 31 = 79
>
> This sequence 19, 23, 31, 43, 47, 59, 71, 79, ...
> is not in OEIS.
>
> Hence it was not arbitrary to first look at sums of three Mersenne
> Primes, and the subsequence of primes within that.
>
> I neglected to mention the analogy to:
> A155877         Sums of three Fermat numbers.
> A166484         Prime sums of three Fermat numbers.
>
> On Thu, Mar 4, 2010 at 4:05 PM, Jonathan Post <jvospost3 at gmail.com> wrote:
>> This was crudely drafted by hand.  But if I did it right, it's not in OEIS.
>>
>> Sums of three Mersenne primes = A000668(i) + A000668(j) + A000668(k),
>> i,j,k integers not necessarily distinct
>>
>> 9, 13, 17, 21, 37, 41, 45, 65, 69, 93, 133, 137, 141, 161, 165, 189,
>> 257, 261, 285, 381, 8197, 8201, 8225, ...
>>
>> Prime sums of three Mersenne primes = A000040 INTERSECTION {A000668(i)
>> + A000668(j) + A000668(k), i,j,k integers not necessarily distinct}
>>
>> 13, 17, 37, 41, 137, 257, ...
>>
>> The first of these seqs is a subset of sums of three odd primes; the
>> second of these is a subset of prime sums of three primes -- but where
>> are these in OEIS?
>>
>> I also would xref A066615  Numbers which are not the sum of two or
>> three distinct primes.
>>
>> Would anyone like to correct me, ore extend these?
>>
>> And are there any strong feelings as to whether these are worth
>> submitting or not?
>>
>> Thank you,
>>
>> Jonathan Vos Post
>>
>
>
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```