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A174056 Prime sums of three Mersenne primes. Primes in A174055.

%I A174056 #16 Sep 22 2020 13:47:00
%S A174056 13,17,37,41,137,257,2147483777,162259895799233006081715459850241
%N A174056 Prime sums of three Mersenne primes. Primes in A174055.
%C A174056 Sums of five Mersenne primes can also be prime (though, obviously sums of an even number of Mersenne primes are even).
%C A174056 3 + 3 + 3 + 3 + 7 = 19
%C A174056 3 + 3 + 3 + 7 + 7 = 23
%C A174056 3 + 7 + 7 + 7 + 7 = 31
%C A174056 3 + 3 + 3 + 3 + 31 = 43
%C A174056 3 + 3 + 3 + 7 + 31 = 47
%C A174056 7 + 7 + 7 + 7 + 31 = 59
%C A174056 3 + 3 + 3 + 31 + 31 = 71
%C A174056 3 + 7 + 7+ 31 + 31 = 79
%C A174056 That sequence of sums of five Mersenne primes 19, 23, 31, 43, 47, 59, 71, 79, ... is A269666.
%C A174056 No other terms < 10^1000. Conjecture: these are all the terms. - _Robert Israel_, Mar 02 2016
%F A174056 A000668(i) + A000668(j) + A000668(k), with integers i,j,k not necessarily distinct. The supersequence of sums of three Mersenne primes is A174055.
%e A174056 a(1) = 3 + 3 + 7 = 13. a(2) = 3 + 7 + 7 = 17. a(3) = 3 + 3 + 31 = 37. a(4) = 3 + 7 + 31 = 41. a(5) = 3 + 7 + 127 = 137. a(6) = 3 + 127 + 127 = 257.
%p A174056 N:= 10^1000: # to get all terms <= N
%p A174056 for n from 1 while numtheory:-mersenne([n]) < N do od:
%p A174056 S:= {seq(numtheory:-mersenne([i]),i=1..n-1)}:
%p A174056 sort(select(isprime,convert(select(`<=`,{seq(seq(seq(s+t+u,s=S),t=S),u=S)},N),list))); # _Robert Israel_, Mar 02 2016
%t A174056 Select[Total/@Tuples[Table[2^MersennePrimeExponent[n]-1,{n,20}],3],PrimeQ]//Union (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 22 2020 *)
%Y A174056 Cf. A000668, A171251, A171252, A171253, A171254, A171255, A174055, A174057, A269666.
%Y A174056 Cf. A155877 (sums of three Fermat numbers).
%Y A174056 Cf. A166484 (prime sums of three Fermat numbers).
%K A174056 more,nonn
%O A174056 1,1
%A A174056 _Jonathan Vos Post_, Mar 06 2010
%E A174056 a(7)-a(8) from _Donovan Johnson_, Dec 22 2010