A334065 - OEIS

%I #12 Apr 27 2020 13:07:44

%S 0,0,0,8,20,12,56,128,108,200,308,312,416,336,420,512,1088,1080,456,

%T 1160,1512,1892,2024,2928,2900,2028,3456,2744,3132,4320,2480,6464,

%U 6732,2040,6440,7776,6956,8588,11388,6720,8036,13272,11180,7392,16920,10856,10152,16032

%N Total area of all triangles such that p + q = 2*n, p < q (p, q prime), with base (q + p) and height (q - p).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>

%H <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = 2*n * Sum_{i=1..n} (n-i) * c(i) * c(2*n-i), where c is the prime characteristic (A010051).

%e a(4) = 8; 2*4 = 8 has one Goldbach partition: (5,3). The area of the triangle is (5 + 3)*(5 - 3)/2 = 8.

%e a(8) = 128; 2*8 = 16 has the two Goldbach partitions: (13,3) and (11,5). The total area of the two triangles is (13 + 3)*(13 - 3)/2 + (11 + 5)*(11 - 5)/2 = 80 + 48 = 128.

%t Table[2 n*Sum[(n - i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[2 n - i] - PrimePi[2 n - i - 1]), {i, n}], {n, 80}]

%o (PARI) a(n) = my(s=0); forprime(p=1, n, if (isprime(2*n-p), s+=n-p)); 2*n*s; \\ _Michel Marcus_, Apr 14 2020

%Y Cf. A010051.

%K nonn,easy

%O 1,4

%A _Wesley Ivan Hurt_, Apr 13 2020