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

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, 26, 26, 32, 33, 35, 36, 39, 40, 41, 43, 44, 46, 48, 49, 52, 53, 53, 54, 58, 63, 64, 65, 65, 67, 68, 71, 73, 75, 77, 78, 79, 80, 81, 84, 90, 91, 92, 92, 98, 100, 104, 105, 105, 107, 110, 112, 114

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

• J. Stauduhar, Table of n, a(n) for n = 1..5000

Crossrefs

Subsequence of A224708.

Cf. A000040, A066247.

Programs

Mathematica

z = 400; c = Select[Range[2, z], ! PrimeQ@# &];  (* A002808 *)
d = Select[Range[2, z], ! PrimeQ@# && OddQ@# &];  (* A014076 *)
a[n_] := Length[Intersection[c, Prime[n] - Select[d, # < Prime[n] &]]];
Table[a[n], {n, 1, 120}] (* A024683 *)
(* Clark Kimberling, Jul 21 2020 *)
Table[Count[IntegerPartitions[Prime[n],{2}],?(AllTrue[#,CompositeQ]&&#[[1]]!= #[[2]]&)],{n,80}] (* _Harvey P. Dale, Feb 24 2023 *)

Formula

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