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.

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, 44, 45, 46, 47, 49, 51, 53, 55, 58, 61, 62, 63, 64, 65, 66, 72, 74, 75, 76, 77, 78, 79, 80, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 101, 102

Offset: 1

Author

Zhi-Wei Sun, Oct 07 2013

Keywords

• nonn

Comments

This sequence is motivated by the author's conjecture in the comments in A230040.

Conjecture: a(n) < 2*n for all n > 2.

Examples

			a(1) = 3 since 3 = 1 + 1 + 1, and 6*1-1=5 is a Sophie Germain prime.
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.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..2000

Crossrefs

Cf. A005384, A230040.

Programs

Mathematica

m=0
SQ[n_]:=SQ[n]=PrimeQ[n]&&PrimeQ[2n+1]
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],
m=m+1;Print[m," ",n];Goto[aa]],{i,1,n/3},{j,i,(n-i)/2}];
Label[aa];Continue,{n,1,102}]
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 *)