cp's OEIS Frontend

A294345 Sum of the products of the smaller and larger parts of the Goldbach partitions of n into two distinct parts.

%I A294345 #35 Feb 16 2025 08:33:51
%S A294345 0,0,0,0,6,0,10,15,14,21,0,35,22,33,26,94,0,142,34,142,38,142,0,357,
%T A294345 46,202,0,302,0,591,58,334,62,491,0,980,0,217,74,821,0,1340,82,785,86,
%U A294345 827,0,1987,94,1512,0,1353,0,2677,106,1421,0,1479,0,4242,118
%N A294345 Sum of the products of the smaller and larger parts of the Goldbach partitions of n into two distinct parts.
%C A294345 Sum of the areas of the distinct rectangles with prime length and width such that L + W = n, W < L. For example, a(14) = 33; the only rectangle is 3 X 11 and 3*11 = 33 (the 7 X 7 rectangle is not considered since we have W < L).
%H A294345 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GoldbachPartition.html">Goldbach Partition</a>
%H A294345 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goldbach%27s_conjecture">Goldbach's conjecture</a>
%H A294345 <a href="/index/Go#Goldbach">Index entries for sequences related to Goldbach conjecture</a>
%H A294345 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A294345 a(n) = Sum_{i=2..floor((n-1)/2)} i * (n-i) * c(i) * c(n-i), where c is the prime characteristic (A010051).
%F A294345 a(k) = 0, for k in A166081. - _Michel Marcus_, Oct 30 2017
%e A294345 a(22) = 142; the Goldbach partitions of 22 are (19,3), (17,5) and (11,11) (we do not consider (11,11) since we only count prime parts which are distinct). Then the sum of the products of the smaller and larger parts from each partition is 19*3 + 17*5 = 142.
%p A294345 with(numtheory): A294345:=n->add(i*(n-i)*(pi(i)-pi(i-1))*(pi(n-i)-pi(n-i-1)), i=2..floor((n-1)/2)): seq(A294345(n), n=1..100);
%t A294345 Table[Sum[i (n - i) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i] - PrimePi[n - i - 1]), {i, 2, Floor[(n-1)/2]}], {n, 60}]
%t A294345 Table[Total[Times@@@Select[IntegerPartitions[n,{2}],AllTrue[#,PrimeQ] && #[[1]]!=#[[2]]&]],{n,70}] (* _Harvey P. Dale_, Jul 29 2021 *)
%o A294345 (PARI) a(n) = sum(i=1, (n-1)\2, i*isprime(i)*(n-i)*isprime(n-i)); \\ _Michel Marcus_, Nov 08 2017
%Y A294345 Cf. A010051, A166081, A243485.
%K A294345 nonn,easy
%O A294345 1,5
%A A294345 _Wesley Ivan Hurt_, Oct 28 2017