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

1,1

Comments

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

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

Links

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

Example

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.

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 *)

Crossrefs

Cf. A005384, A230040.

Sequence in context: A285659 A256421 A253200 * A039096 A083121 A298006

Adjacent sequences: A227935 A227936 A227937 * A227939 A227940 A227941

Keyword

nonn

Author

Zhi-Wei Sun, Oct 07 2013

Status

approved