login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A071538
Number of twin prime pairs (p, p+2) with p <= n.
19
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
1,5
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
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