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%I #5 Apr 01 2015 22:13:40
%S 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,
%T 9,12,11,10,12,8,10,10,10,9,9,13,11,10,13,11,11,12,10,10,14,14,12,12,
%U 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
%N Number of ways to write n as the sum of two (unordered) distinct elements of the set {floor(p/3): p is prime}.
%C 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}.
%C 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.
%H Zhi-Wei Sun, <a href="/A256555/b256555.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%t S[n_]:=Union[Table[Floor[Prime[k]/3], {k,1,PrimePi[3n+2]}]]
%t L[n_]:=Length[S[n]]
%t Do[r=0;Do[If[Part[S[n],x]>=n/2,Goto[cc]];
%t 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}]
%Y Cf. A000040, A002375, A256544.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Apr 01 2015
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