A062602 Number of ways of writing n = p+c with p prime and c nonprime (1 or a composite number).

0, 0, 1, 1, 0, 2, 1, 2, 2, 1, 4, 3, 3, 3, 4, 2, 6, 3, 5, 4, 6, 3, 8, 3, 7, 4, 9, 5, 9, 4, 8, 7, 9, 4, 11, 3, 11, 9, 10, 6, 12, 5, 11, 8, 12, 7, 14, 5, 13, 7, 15, 9, 15, 6, 14, 10, 16, 9, 16, 5, 15, 13, 16, 8, 18, 6, 18, 15, 17, 9, 19, 8, 18, 12, 19, 11, 21, 7, 21, 14, 20, 13, 22, 7, 21, 14

Offset: 1

Author

Labos Elemer, Jul 04 2001

Keywords

• nonn
• easy

Examples

			n = 22 has floor(n/2) = 11 partitions of form n = a + b; 3 partitions are of prime + prime [3 + 19 = 5 + 17 = 11 + 11], 3 partitions are of prime + nonprime [2 + 20 = 7 + 15 = 13 + 9], 5 partitions are nonprime + nonprime [1 + 21 = 4 + 18 = 6 + 16 = 8 + 14 = 10 + 12]. So a(22) = 3.
		

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A061358, A014092, A062610, A224712.

Programs

Mathematica

Table[Length[Select[Range[Floor[n/2]], (PrimeQ[#] && Not[PrimeQ[n - #]]) || (Not[PrimeQ[#]] && PrimeQ[n - #]) &]], {n, 80}] (* Alonso del Arte, Apr 21 2013 *)
Table[Length[Select[IntegerPartitions[n,{2}],AnyTrue[#,PrimeQ] && !AllTrue[ #,PrimeQ]&]],{n,90}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 19 2020 *)

Formula

a(n+1) = SUM(A010051(k)*A005171(n-k+1): 1<=k<=n). [From Reinhard Zumkeller, Nov 05 2009]

a(n) + A061358(n) + A062610(n) = A004526(n). - R. J. Mathar, Sep 10 2021