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Number of primes <= n that are safe primes or Sophie Germain primes.
6

%I #10 Nov 06 2022 07:48:29

%S 0,1,2,2,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,

%T 7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,11,11,11,11,11,

%U 11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,12,12,12,12,12

%N Number of primes <= n that are safe primes or Sophie Germain primes.

%H Reinhard Zumkeller, <a href="/A156876/b156876.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A156874(n)+A156875(n)-A156877(n) = A000720(n)-A156878(n).

%e a(120) = #{2,3,5,7,11,23,29,41,47,53,59,83,89,107,113} = 15.

%t Accumulate[Table[If[AllTrue[{n,2n+1},PrimeQ]||AllTrue[{n,(n-1)/2}, PrimeQ],1,0],{n,100}]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Sep 24 2019 *)

%o (PARI) a(n) = my(nb=0); forprime(p=2, n, if (isprime(2*p+1) || isprime((p-1)/2), nb++)); nb; \\ _Michel Marcus_, Nov 06 2022

%Y Cf. A156658, A005384, A005385, A000720.

%Y Cf. A156874, A156875, A156877, A156878.

%K nonn

%O 1,3

%A _Reinhard Zumkeller_, Feb 18 2009