cp's OEIS Frontend

A301776 Prime numbers p with the property that all even numbers n (2 < n <= 2p) are the sum of two primes <= p.

%I A301776 #29 Aug 06 2018 05:31:22
%S A301776 2,3,5,7,13,19,109
%N A301776 Prime numbers p with the property that all even numbers n (2 < n <= 2p) are the sum of two primes <= p.
%C A301776 Conjecture: this sequence is finite - it has 7 terms only. Conjecture verified up to first 10^5 primes.
%C A301776 This sequence is related to the Goldbach Strong Conjecture.
%H A301776 Marcin Barylski, <a href="/A301776/a301776.cpp.txt">C++ program</a>
%H A301776 Marcin Barylski, <a href="/A301776/a301776.png">Sum building from first primes vs. theoretical maximum</a> - the first 200 rounds of the algorithm. If red and green lines ever touch each other, we have a new term.
%H A301776 Marcin Barylski, <a href="http://tas-moto.org/research/GoldbachSumPrimes.pdf">Goldbach Strong Conjecture Verification Using Prime Numbers</a>
%e A301776 a(1)=2 because all even numbers 2 < n <= 2*2 (there is just one such number: 4) can be expressed as a sum of 2 only: 4=2+2.
%e A301776 a(2)=3 because 4=2+2, 6=3+3.
%e A301776 a(3)=5 because 4=2+2, 6=3+3, 8=5+3, 10=5+5.
%e A301776 a(4)=7 because 4=2+2, 6=3+3, 8=5+3, 10=5+5, 12=5+7, 14=7+7.
%e A301776 a(5)=13 (and is not 11) because 20 cannot be expressed as a sum of two primes from a set {2,3,5,7,11} but all even numbers 2 < n <= 26 can be expressed as a sum of two primes from a set {2,3,5,7,11,13}.
%t A301776 Select[Prime@ Range[500], Function[p, SameQ[Select[Union@ Map[Total, Tuples[Prime@ Range@ PrimePi@ p, 2]], And[EvenQ@ #, # > p] &], Range[p + 1 + Boole@ EvenQ@ p, 2 p, 2]]]] (* _Michael De Vlieger_, Apr 10 2018 *)
%o A301776 (C++) See Barylski link.
%o A301776 (PARI) isok(p) = {vp = primes(primepi(p)); slist = List(); for (i=1, #vp, for (j=1, i, if (!((vp[i]+vp[j]) % 2), listput(slist, vp[i]+vp[j])););); #Set(slist) == (p-1);}
%o A301776 lista(nn) = forprime(p=2, nn, if (isok(p), print1(p, ", "))); \\ _Michel Marcus_, Apr 09 2018
%Y A301776 Cf. A002372 (number of ordered Goldbach partitions).
%K A301776 nonn,more
%O A301776 1,1
%A A301776 _Marcin Barylski_, Mar 26 2018