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Number of twin primes between p and p^2 (inclusive) where p is the n-th prime.
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%I #31 Feb 29 2020 16:17:41

%S 1,3,6,9,16,19,32,35,42,58,61,82,96,101,122,148,174,183,220,242,247,

%T 276,304,332,374,404,417,436,447,468,552,576,630,641,730,749,788,822,

%U 864,910,960,985,1082,1095,1134,1149,1252,1370,1416,1433,1464,1528,1545,1636,1702

%N Number of twin primes between p and p^2 (inclusive) where p is the n-th prime.

%C Similar sequences given in cross-references have further information and references; in particular A273257 has much more efficient PARI code. - _M. F. Hasler_, Jun 27 2019

%H Chai Wah Wu, <a href="/A308777/b308777.txt">Table of n, a(n) for n = 1..7944</a>

%H Jon S. Birdsey, Geza Schay, <a href="https://arxiv.org/abs/1906.09220">A Sieve for Twin Primes</a>, arXiv:1906.09220 [math.NT], 2019.

%e There is a single twin prime (3) between 2 and 4, so a(1) = 1.

%e There are 3 twin primes (3, 5 and 7) between 3 and 9, so a(2) = 3.

%p a:= n-> (p-> add(`if`(isprime(j) and (isprime(j-2) or

%p isprime(j+2)), 1, 0), j=p..p^2))(ithprime(n)):

%p seq(a(n), n=1..55); # _Alois P. Heinz_, Jun 25 2019

%t a[n_] := With[{p = Prime[n]}, Sum[Boole[PrimeQ[k] && (PrimeQ[k-2] || PrimeQ[k+2])], {k, p, p^2}]];

%t Array[a, 55] (* _Jean-François Alcover_, Feb 29 2020 *)

%o (PARI) a(n) = my(p=prime(n)); sum(k=p, p^2, isprime(k) && (isprime(k-2) || isprime(k+2)));

%o (Python)

%o from sympy import prime, prevprime, nextprime

%o def A308777(n):

%o if n == 1:

%o return 1

%o c, p = 0, prime(n)

%o p2, x = p**2, [prevprime(p), p , nextprime(p)]

%o while x[1] <= p2:

%o if x[1] - x[0] == 2 or x[2] - x[1] == 2:

%o c += 1

%o x = x[1:] + [nextprime(x[2])]

%o return c # _Chai Wah Wu_, Jun 25 2019

%Y Cf. A001097 (twin primes), A054272, A057767 (twin pairs between p(n)^2 and p(n+1)^2), A088019.

%Y Cf. A143738 (twin primes between n and n^2), A273257 (twin pairs between prime(n) and prime(n)^2).

%K nonn

%O 1,2

%A _Michel Marcus_, Jun 24 2019