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A071538
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Number of twin prime pairs (p, p+2) with p <= n.
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19
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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
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OFFSET
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1,5
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COMMENTS
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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
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REFERENCES
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S. Lang, The Beauty of Doing Mathematics, pp. 12-15; 21-22, Springer-Verlag NY 1985.
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LINKS
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EXAMPLE
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a(30) = 5, since (29,31) is included along with (3,5), (5,7), (11,13) and (17,19).
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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