cp's OEIS Frontend

A278700 Number of Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.

%I A278700 #33 Feb 16 2025 08:33:37
%S A278700 0,0,1,1,2,1,1,0,2,0,1,3,1,0,3,0,1,2,1,0,1,0,1,0,0,1,0,0,1,5,1,0,0,1,
%T A278700 0,0,1,0,1,0,1,3,1,0,1,0,1,0,0,1,0,0,2,0,0,1,0,0,1,2,1,0,0,1,0,0,1,0,
%U A278700 1,0,1,1,1,0,0,1,0,0,1,0,2,0,1,0,0,1,0,0,1
%N A278700 Number of Goldbach partitions (p,q) of 2n such that all primes from p to q (inclusive) appear as a part in some Goldbach partition of p+q = 2n.
%C A278700 Records are a(3) = 1, a(5) = 2, a(12) = 3, a(30) = 5, a(165) = 6, a(8021811) = 7. - _Charles R Greathouse IV_, Nov 30 2016
%C A278700 a(n) <= A002375(n). - _Wesley Ivan Hurt_, Dec 17 2016
%H A278700 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A278700 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A278700 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A278700 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A278700 a(n) = Sum_{i=3..n} (c(i) * c(2*n-i) * Product_{k=i..n} (1 - abs(c(k) - c(2*n-k)))), where c is the prime characteristic (A010051).
%e A278700 a(5) = 2; There are 2 Goldbach partitions of 2*5 = 10, namely (3,7) and (5,5). (3,7) satisfies the condition that every prime from 3 to 7 (inclusive) appears as a part in some Goldbach partition of 10. This is also true of (5,5) since 5 appears in its own partition.
%t A278700 Table[Sum[Times @@ Map[Boole@ PrimeQ@ # &, {i, 2 n - i}] * Product[(1 - Abs[Subtract @@ Map[Boole@ PrimeQ@ # &, {k, 2 n - k}]]), {k, i, n}], {i, 3, n}], {n, 89}] (* _Michael De Vlieger_, Nov 30 2016 *)
%o A278700 (PARI) a(n) = sum(i=3, n, (ispseudoprime(i) * ispseudoprime(2*n-i) * prod(k=i, n, (1-abs(ispseudoprime(k)-ispseudoprime(2*n-k)))))) \\ _Felix Fröhlich_, Nov 28 2016
%o A278700 (PARI) a(n) = if(n<3, return(0)); my(s,p=n,N=2*n); forprime(q=n, N, while(p<q, if(isprime(N-p), return(s));p++); p++; if(!isprime(N-q), break); s++); s \\ _Charles R Greathouse IV_, Nov 30 2016
%Y A278700 Cf. A000720, A002375, A010051, A055381, A279103.
%K A278700 nonn,easy
%O A278700 1,5
%A A278700 _Wesley Ivan Hurt_, Nov 26 2016