A256555 Number of ways to write n as the sum of two (unordered) distinct elements of the set {floor(p/3): p is prime}.

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

Offset: 1

Author

Zhi-Wei Sun, Apr 01 2015

Keywords

• nonn

Comments

Conjecture: For any integer m > 2, every positive integer can be written as the sum of two distinct elements of the set {floor(p/m): p is prime}.

Note that Goldbach's conjecture essentially asserts that any integer n > 1 can be written as floor(p/2) + floor(q/2) with p and q prime.

Examples

			 a(4) = 2 since 4 = 0 + 4 = 1 + 3 with 0,1,3,4 elements of the set {floor(p/3): p is prime}. Note that floor(2/3) = 0, floor(3/3) = 1, floor(11/3) = 3 and floor(13/3) = 4.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A002375, A256544.

Programs

Mathematica

S[n_]:=Union[Table[Floor[Prime[k]/3], {k,1,PrimePi[3n+2]}]]
L[n_]:=Length[S[n]]
Do[r=0;Do[If[Part[S[n],x]>=n/2,Goto[cc]];
If[MemberQ[S[n], n-Part[S[n],x]]==True,r=r+1]; Continue,{x,1,L[n]}];Label[cc];Print[n," ",r];Continue, {n,1,100}]