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

%I A071538 #32 Feb 16 2025 08:32:46
%S A071538 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,
%T A071538 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,
%U A071538 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
%N A071538 Number of twin prime pairs (p, p+2) with p <= n.
%C A071538 The convention is followed that a twin prime is <= n if its smaller member is <= n.
%C A071538 Except for (3, 5), every pair of twin primes is congruent (-1, +1) (mod 6). - _Daniel Forgues_, Aug 05 2009
%C A071538 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
%D A071538 S. Lang, The Beauty of Doing Mathematics, pp. 12-15; 21-22, Springer-Verlag NY 1985.
%H A071538 Daniel Forgues, <a href="/A071538/b071538.txt">Table of n, a(n) for n = 1..99998</a>
%H A071538 Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/">Some Results of Computational Research in Prime Numbers</a>.
%H A071538 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TwinPrimes.html">Twin Primes</a>.
%e A071538 a(30) = 5, since (29,31) is included along with (3,5), (5,7), (11,13) and (17,19).
%t A071538 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 *)
%o A071538 (PARI) A071538(n) = local(s=0,L=0); forprime(p=3,n+2,L==p-2 & s++; L=p); s
%o A071538 /* 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 }
%o A071538 /* The following gives a reasonably good estimate for small and for large values of n (cf. A007508): */
%o A071538 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
%Y A071538 Cf. A007508, A033843, A001359, A006512.
%K A071538 nonn
%O A071538 1,5
%A A071538 _Reinhard Zumkeller_, May 30 2002
%E A071538 Definition edited by _Daniel Forgues_, Jul 29 2009