OFFSET
1,5
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
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.
EXAMPLE
a(30) = 5, since (29,31) is included along with (3,5), (5,7), (11,13) and (17,19).
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, May 30 2002
EXTENSIONS
Definition edited by Daniel Forgues, Jul 29 2009
STATUS
approved