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A024683 a(n) is the number of ways prime(n) is a sum of two composite numbers r,s satisfying r < s.

%I A024683 #18 Feb 24 2023 10:13:31
%S A024683 0,0,0,0,0,1,1,2,2,4,5,6,7,8,8,10,12,13,14,15,16,17,18,20,23,24,25,25,
%T A024683 26,26,32,33,35,36,39,40,41,43,44,46,48,49,52,53,53,54,58,63,64,65,65,
%U A024683 67,68,71,73,75,77,78,79,80,81,84,90,91,92,92,98,100,104,105,105,107,110,112,114
%N A024683 a(n) is the number of ways prime(n) is a sum of two composite numbers r,s satisfying r < s.
%H A024683 J. Stauduhar, <a href="/A024683/b024683.txt">Table of n, a(n) for n = 1..5000</a>
%F A024683 a(n) = Sum_{i=4..floor((prime(n)-1)/2)} c(i) * c(prime(n)-i), where c is the characteristic function of composite numbers (A066247) and prime(n) is the n-th prime (A000040). - _Wesley Ivan Hurt_, Sep 08 2020
%t A024683 z = 400; c = Select[Range[2, z], ! PrimeQ@# &];  (* A002808 *)
%t A024683 d = Select[Range[2, z], ! PrimeQ@# && OddQ@# &];  (* A014076 *)
%t A024683 a[n_] := Length[Intersection[c, Prime[n] - Select[d, # < Prime[n] &]]];
%t A024683 Table[a[n], {n, 1, 120}] (* A024683 *)
%t A024683 (* _Clark Kimberling_, Jul 21 2020 *)
%t A024683 Table[Count[IntegerPartitions[Prime[n],{2}],_?(AllTrue[#,CompositeQ]&&#[[1]]!= #[[2]]&)],{n,80}] (* _Harvey P. Dale_, Feb 24 2023 *)
%Y A024683 Subsequence of A224708.
%Y A024683 Cf. A000040, A066247.
%K A024683 nonn,easy
%O A024683 1,8
%A A024683 _Clark Kimberling_