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%I #5 Jun 24 2014 10:27:46
%S 4937487,7763129,8572423,8770977,14024923,15515130,18297902,18935975,
%T 20755826,20986868,22661767,25060321,26606907,28884783,29283671
%N Positive integers n such that all those prime(n+i) + prime(n+j) (i,j = 0..7) are squarefree.
%C Conjecture: For any integer m > 0, there are infinitely many positive integers n such that all those prime(n+i) + prime(n+j) (i,j = 0, ..., m) are squarefree.
%e a(1) = 4937487, and the 8 consecutive primes prime(4937487+i) (i = 0..7) have the values 84885631, 84885643, 84885667, 84885679, 84885727, 84885739, 84885751,84885763 respectively. The sum of any two of the 8 consecutive primes is squarefree.
%t SFQ[n_]:=SquareFreeQ[n]
%t m=0;LL[n_]:=Sum[Boole[Mod[Prime[n+i],4]==1],{i,0,7}]
%t Do[If[LL[n]>0&&LL[n]<8,Goto[aa]];Do[If[SFQ[Prime[n+i]+Prime[n+j]]==False,Goto[aa]],{j,1,7},{i,0,j-1}];m=m+1;Print[m," ",n];Label[aa];Continue,{n,1,29283671}]
%Y Cf. A000040, A005117, A244254, A244264.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Jun 24 2014
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