A088019 -id:A088019 - cp's OEIS frontend

A088018 Number of twin-prime pairs between n and 2n (inclusive).

0, 0, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 7, 6, 6, 6, 6

Offset: 1

T. D. Noe, Sep 18 2003, Feb 17 2011

To be counted, both members of the twin-prime pair must be between n and 2n, inclusive. It appears that a(n) > 0 for all n > 6. However, it has not been proved that there are an infinite number of twin primes.

Same as the number of lower twin primes between n-1 and 2(n-1), exclusive. If the twin prime conjecture is true, there are at least n lower twin primes between x/2 and x for all x >= A186312(n).

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Twin Primes

Cf. A035250 (number of primes between n and 2n), A088019 (number of twin primes between n and 2n).

nn=100; p=Select[Prime[Range[PrimePi[2*nn]]], PrimeQ[#+2] &]; t=Table[0, {nn}]; Do[t[[Span[Ceiling[i/2], Min[nn,i-1]]]]++, {i, p}]; Prepend[t,0]
Table[Total[Length /@ Split[Select[Range[n, 2 n], PrimeQ], #2 - #1 == 2 &] - 1], {n, 105}] (* Jayanta Basu, Aug 12 2013 *)

A308777 Number of twin primes between p and p^2 (inclusive) where p is the n-th prime.

Original entry on oeis.org

1, 3, 6, 9, 16, 19, 32, 35, 42, 58, 61, 82, 96, 101, 122, 148, 174, 183, 220, 242, 247, 276, 304, 332, 374, 404, 417, 436, 447, 468, 552, 576, 630, 641, 730, 749, 788, 822, 864, 910, 960, 985, 1082, 1095, 1134, 1149, 1252, 1370, 1416, 1433, 1464, 1528, 1545, 1636, 1702

Offset: 1

Michel Marcus, Jun 24 2019

nonn

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

			There is a single twin prime (3) between 2 and 4, so a(1) = 1.
There are 3 twin primes (3, 5 and 7) between 3 and 9, so a(2) = 3.

Chai Wah Wu, Table of n, a(n) for n = 1..7944
Jon S. Birdsey, Geza Schay, A Sieve for Twin Primes, arXiv:1906.09220 [math.NT], 2019.

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

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

a:= n-> (p-> add(`if`(isprime(j) and (isprime(j-2) or
        isprime(j+2)), 1, 0), j=p..p^2))(ithprime(n)):
seq(a(n), n=1..55);  # Alois P. Heinz, Jun 25 2019

a[n_] := With[{p = Prime[n]}, Sum[Boole[PrimeQ[k] && (PrimeQ[k-2] || PrimeQ[k+2])], {k, p, p^2}]];
Array[a, 55] (* Jean-François Alcover, Feb 29 2020 *)

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

from sympy import prime, prevprime, nextprime
def A308777(n):
    if n == 1:
        return 1
    c, p = 0, prime(n)
    p2, x = p**2, [prevprime(p), p , nextprime(p)]
    while x[1] <= p2:
        if x[1] - x[0] == 2 or x[2] - x[1] == 2:
            c += 1
        x = x[1:] + [nextprime(x[2])]
    return c # Chai Wah Wu, Jun 25 2019

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A088018 Number of twin-prime pairs between n and 2n (inclusive).

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A308777 Number of twin primes between p and p^2 (inclusive) where p is the n-th prime.

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