cp's OEIS Frontend

A226237 Sum of the parts in the Goldbach partitions of 2n.

%I A226237 #30 Feb 16 2025 08:33:19
%S A226237 0,4,6,8,20,12,28,32,36,40,66,72,78,56,90,64,136,144,76,120,168,132,
%T A226237 184,240,200,156,270,168,232,360,186,320,396,136,350,432,370,380,546,
%U A226237 320,410,672,430,352,810,368,470,672,294,600,816,520,636,864,660,784
%N A226237 Sum of the parts in the Goldbach partitions of 2n.
%C A226237 Goldbach's Conjecture states that every positive even integer > 4 is expressible as the sum of two odd primes in at least one way.  This is logically equivalent to the statement that a(n) > 0 for n > 2.
%C A226237 The sum of the parts in the partitions of 2n into exactly two prime parts.
%H A226237 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A226237 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A226237 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A226237 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A226237 a(n) = 2n * A045917(n). a(n) = A185297(n) + A187129(n), n>1.
%e A226237 a(13) = 78.  Since 2*13 = 26 has exactly 3 Goldbach partitions: (23,3),(19,7), and (13,13).  The sum of the parts gives: 23+19+13+13+7+3 = 78.
%p A226237 with(numtheory); A226237:=n->2*n*sum( (pi(i)-pi(i-1)) * (pi(2*n-i)-pi(2*n-i-1)), i=1..n); seq(A226237(n), n=1..100);
%t A226237 Table[ 2 n*Sum[ Floor[2/PrimeOmega[2 n*i - i^2]], {i, 2, n}], {n,
%t A226237   100}]
%Y A226237 Cf. A045917, A185297, A187129, A187619 (Sum of differences).
%K A226237 nonn,easy
%O A226237 1,2
%A A226237 _Wesley Ivan Hurt_, Aug 25 2013