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

1,6

Links

Donovan Johnson, Table of n, a(n) for n = 1..10000

Index entries for sequences related to Goldbach conjecture

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

Example

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.

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 *)

Crossrefs

Cf. A061358, A014092, A062610, A224712.

Sequence in context: A242210 A035369 A129719 * A123148 A173410 A166548

Adjacent sequences: A062599 A062600 A062601 * A062603 A062604 A062605

Keyword

nonn,easy

Author

Labos Elemer, Jul 04 2001

Status

approved