A071538 Number of twin prime pairs (p, p+2) with p <= n.

0, 0, 1, 1, 2, 2, 2, 2, 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, 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, 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

Offset: 1

Author

Reinhard Zumkeller, May 30 2002

Keywords

• nonn

Comments

The convention is followed that a twin prime is <= n if its smaller member is <= n.

Except for (3, 5), every pair of twin primes is congruent (-1, +1) (mod 6). - Daniel Forgues, Aug 05 2009

This function is sometimes known as pi_2(n). If this name is used, there is no obvious generalization for pi_k(n) for k > 2. - Franklin T. Adams-Watters, Jun 01 2014

Examples

			a(30) = 5, since (29,31) is included along with (3,5), (5,7), (11,13) and (17,19).
		

References

• S. Lang, The Beauty of Doing Mathematics, pp. 12-15; 21-22, Springer-Verlag NY 1985.

Links

• Daniel Forgues, Table of n, a(n) for n = 1..99998
• Thomas R. Nicely, Some Results of Computational Research in Prime Numbers.
• Eric Weisstein's World of Mathematics, Twin Primes.

Crossrefs

Cf. A007508, A033843, A001359, A006512.

Programs

Mathematica

primePi2[1] = 0; primePi2[n_] := primePi2[n] = primePi2[n - 1] + Boole[PrimeQ[n] && PrimeQ[n + 2]]; Table[primePi2[n], {n, 100}] (* T. D. Noe, May 23 2013 *)

PARI

A071538(n) = local(s=0,L=0); forprime(p=3,n+2,L==p-2 & s++; L=p); s
/* For n > primelimit, one may use: */ A071538(n) = { local(s=isprime(2+n=precprime(n))&n,L); while( n=precprime(L=n-2),L==n & s++); s }
/* The following gives a reasonably good estimate for small and for large values of n (cf. A007508): */
A071538est(n) = 1.320323631693739*intnum(t=2,n+1/n,1/log(t)^2)-log(n) /* (The constant 1.320... is A114907.) */ \\ M. F. Hasler, Dec 10 2008

Extensions

Definition edited by Daniel Forgues, Jul 29 2009