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A227938 List of those numbers which can be written as x + y + z (x, y, z > 0) such that all the six numbers 6*x-1, 6*y-1, 6*z-1, 6*x*y-1, 6*x*z-1 and 6*y*z-1 are Sophie Germain primes.

%I A227938 #15 Jul 23 2024 14:31:19
%S A227938 3,4,5,6,7,9,10,11,15,16,17,18,19,20,21,24,25,28,31,32,33,34,35,41,42,
%T A227938 44,45,46,47,49,51,53,55,58,61,62,63,64,65,66,72,74,75,76,77,78,79,80,
%U A227938 84,86,87,88,89,90,91,92,93,94,101,102
%N A227938 List of those numbers which can be written as x + y + z (x, y, z > 0) such that all the six numbers 6*x-1, 6*y-1, 6*z-1, 6*x*y-1, 6*x*z-1 and 6*y*z-1 are Sophie Germain primes.
%C A227938 This sequence is motivated by the author's conjecture in the comments in A230040.
%C A227938 Conjecture: a(n) < 2*n for all n > 2.
%H A227938 Zhi-Wei Sun, <a href="/A227938/b227938.txt">Table of n, a(n) for n = 1..2000</a>
%e A227938 a(1) = 3 since 3 = 1 + 1 + 1, and 6*1-1=5 is a Sophie Germain prime.
%e A227938 a(7) = 10 since 10 = 1 + 2 + 7, and 6*1-1=5, 6*2-1=11, 6*7-1=41, 6*1*2-1=11, 6*1*7-1=41, 6*2*7-1=83 are Sophie Germain primes.
%t A227938 m=0
%t A227938 SQ[n_]:=SQ[n]=PrimeQ[n]&&PrimeQ[2n+1]
%t A227938 Do[Do[If[SQ[6i-1]&&SQ[6j-1]&&SQ[6(n-i-j)-1]&&SQ[6i*j-1]&&SQ[6*i(n-i-j)-1]&&SQ[6*j(n-i-j)-1],
%t A227938 m=m+1;Print[m," ",n];Goto[aa]],{i,1,n/3},{j,i,(n-i)/2}];
%t A227938 Label[aa];Continue,{n,1,102}]
%t A227938 sgpQ[{x_,y_,z_}]:=AllTrue[{6x-1,6y-1,6z-1,6x y-1,6x z-1,6y z-1,2(6x-1)+1,2(6y-1)+1,2(6z-1)+ 1,2(6x y-1)+1,2(6x z-1)+1,2(6y z-1)+1},PrimeQ]; Select[Total/@Select[Tuples[Range[100],3],sgpQ]//Union,#<110&] (* _Harvey P. Dale_, Jul 23 2024 *)
%Y A227938 Cf. A005384, A230040.
%K A227938 nonn
%O A227938 1,1
%A A227938 _Zhi-Wei Sun_, Oct 07 2013