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A243485 Sum of all the products formed by multiplying the corresponding smaller and larger parts of the Goldbach partitions of n.

%I A243485 #38 Feb 16 2025 08:33:22
%S A243485 0,0,0,4,6,9,10,15,14,46,0,35,22,82,26,94,0,142,34,142,38,263,0,357,
%T A243485 46,371,0,302,0,591,58,334,62,780,0,980,0,578,74,821,0,1340,82,785,86,
%U A243485 1356,0,1987,94,1512,0,1353,0,2677,106,1421,0,2320,0,4242,118
%N A243485 Sum of all the products formed by multiplying the corresponding smaller and larger parts of the Goldbach partitions of n.
%C A243485 a(n) is even for odd n.
%C A243485 If Goldbach's conjecture is true, a(n) > 0 for all even n > 2.
%C A243485 Sum of the areas of the distinct rectangles with prime length and width such that L + W = n, W <= L. For example, a(16) = 94; the two rectangles are 3 X 13 and 5 X 11, and the sum of their areas is 3*13 + 5*11 = 94. - _Wesley Ivan Hurt_, Oct 28 2017
%H A243485 Vincenzo Librandi, <a href="/A243485/b243485.txt">Table of n, a(n) for n = 1..1000</a>
%H A243485 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A243485 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A243485 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A243485 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A243485 a(n) = Sum_{i=2..n/2} i*(n-i) * A064911(i*(n-i)).
%F A243485 a(n) = Sum_{i=1..floor(n/2)} i * (n-i) * A010051(i) * A010051(n-i). - _Wesley Ivan Hurt_, Oct 29 2017
%p A243485 with(numtheory): A243485:=n->add(i*(n-i)*(pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=1..floor(n/2)): seq(A243485(n), n=1..100); # _Wesley Ivan Hurt_, Oct 29 2017
%t A243485 Table[Sum[i*(n - i)*Floor[2/PrimeOmega[i (n - i)]], {i, 2, n/2}], {n,
%t A243485   50}]
%Y A243485 Cf. A002372, A002375, A045917, A064911, A117929.
%K A243485 nonn,easy
%O A243485 1,4
%A A243485 _Wesley Ivan Hurt_, Jun 05 2014