A156876 -id:A156876 - cp's OEIS frontend

A156874 Number of Sophie Germain primes <= n.

0, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10

Offset: 1

Reinhard Zumkeller, Feb 18 2009

nonn

a(n) = Sum_{k=1..n} A156660(k).
a(n) = A156875(2*n+1).
Hardy-Littlewood conjecture: a(n) ~ 2*C2*n/(log(n))^2, where C2=0.6601618158... is the twin prime constant (see A005597).
The truth of the above conjecture would imply that there exists an infinity of Sophie Germain primes (which is also conjectured).
a(n) ~ 2*C2*n/(log(n))^2 is also conjectured by Hardy-Littlewood for the number of twin primes <= n.

			a(120) = #{2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113} = 11.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sophie Germain prime
Wikipedia, Sophie Germain prime

A156875, A156876, A156877, A156878, A000720.
Cf. A005384 Sophie Germain primes p: 2p+1 is also prime.
Cf. A092816.

Accumulate[Table[Boole[PrimeQ[n]&&PrimeQ[2n+1]],{n,1,200}]] (* Enrique Pérez Herrero, Apr 26 2012 *)
Accumulate[Table[If[AllTrue[{n,2n+1},PrimeQ],1,0],{n,200}]]

a(10^n)= A092816(n). - Enrique Pérez Herrero, Apr 26 2012

Edited and commented by Daniel Forgues, Jul 31 2009

A156875 Number of safe primes <= n.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Feb 18 2009

nonn

a(n) = SUM(A156659(k): 1<=k<=n);

a(n) = A156874(floor((n-1)\2)).

			a(120) = #{5, 7, 11, 23, 47, 59, 83, 107} = 8.

R. Zumkeller, Table of n, a(n) for n = 1..10000
Wikipedia, Safe prime

A005385, A156874, A156876, A156877, A156878, A000720.

A156877 Number of primes <= n that are safe primes and also Sophie Germain primes.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4

Offset: 1

Reinhard Zumkeller, Feb 18 2009

nonn

			a(120) = #{5, 11, 23, 83} = 4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A059455, A005384, A005385, A000720.

Cf. A156874, A156875, A156876, A156659, A156660.

a(n) = A156874(n) + A156875(n) - A156876(n).

a(n) = Sum_{k=1..n} A156659(k)*A156660(k).

A156878 Number of primes <= n that are neither safe primes and nor Sophie Germain primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11

Offset: 1

Reinhard Zumkeller, Feb 18 2009

nonn

			a(120) = #{13,17,19,31,37,43,61,67,71,73,79,97,10,103,109} = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A059500, A005384, A005385.

Cf. A000720, A156876.

a(n) = A000720(n) - A156876(n).

A156658 Primes p such that also 2*p+1 or (p-1)/2 is prime.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 29, 41, 47, 53, 59, 83, 89, 107, 113, 131, 167, 173, 179, 191, 227, 233, 239, 251, 263, 281, 293, 347, 359, 383, 419, 431, 443, 467, 479, 491, 503, 509, 563, 587, 593, 641, 653, 659, 683, 719, 743, 761, 809, 839, 863, 887, 911, 953, 983, 1013

Offset: 1

Reinhard Zumkeller, Feb 13 2009

nonn

Union of A005384 and A005385;
The intersection of A005384 and A005385 is given by A059455.
A156660(a(n)) + A156659(a(n)) > 0;
primes occurring in Cunningham chains of the first kind.
A156876 gives the number of these numbers <= n. [Reinhard Zumkeller, Feb 18 2009]

Robert Price, Table of n, a(n) for n = 1..14198
Wikipedia, Sophie Germain prime
Wikipedia, Safe prime
Wikipedia, Cunningham chain

Cf. A005384, A005385, A059500, A156659, A156660, A156876.

select(t -> isprime(t) and (isprime(2*t+1) or isprime((t-1)/2)), [2,seq(p,p=3..10000,2)]); # Robert Israel, May 03 2016

Select[Prime@ Range@ 180, PrimeQ[2 # + 1] || PrimeQ[(# - 1)/2] &] (* Michael De Vlieger, Apr 06 2016 *)

lista(nn) = {forprime(p=2, nn, if (isprime(2*p+1) || isprime((p-1)/2), print1(p, ", ")););} \\ Michel Marcus, Apr 06 2016

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A156874 Number of Sophie Germain primes <= n.

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A156875 Number of safe primes <= n.

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A156877 Number of primes <= n that are safe primes and also Sophie Germain primes.

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A156878 Number of primes <= n that are neither safe primes and nor Sophie Germain primes.

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A156658 Primes p such that also 2*p+1 or (p-1)/2 is prime.

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