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

13, 17, 37, 41, 137, 257, 2147483777, 162259895799233006081715459850241

Offset: 1

Author

Jonathan Vos Post, Mar 06 2010

Keywords

• more
• nonn

Comments

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

That sequence of sums of five Mersenne primes 19, 23, 31, 43, 47, 59, 71, 79, ... is A269666.

No other terms < 10^1000. Conjecture: these are all the terms. - Robert Israel, Mar 02 2016

Examples

			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.
		

Crossrefs

Cf. A000668, A171251, A171252, A171253, A171254, A171255, A174055, A174057, A269666.

Cf. A155877 (sums of three Fermat numbers).

Cf. A166484 (prime sums of three Fermat numbers).

Programs

Maple

N:= 10^1000: # to get all terms <= N
for n from 1 while numtheory:-mersenne([n]) < N do od:
S:= {seq(numtheory:-mersenne([i]),i=1..n-1)}:
sort(select(isprime,convert(select(`<=`,{seq(seq(seq(s+t+u,s=S),t=S),u=S)},N),list))); # Robert Israel, Mar 02 2016

Mathematica

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 *)

Formula

A000668(i) + A000668(j) + A000668(k), with integers i,j,k not necessarily distinct. The supersequence of sums of three Mersenne primes is A174055.

Extensions

a(7)-a(8) from Donovan Johnson, Dec 22 2010