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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.
2
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.
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
Sequence in context: A285659 A256421 A253200 * A039096 A083121 A298006
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 07 2013
STATUS
approved