A143738 -id:A143738 - cp's OEIS frontend

A273257 Number of twin primes between prime(n) and prime(n)^2.

0, 1, 3, 4, 8, 9, 16, 17, 21, 29, 30, 41, 48, 50, 61, 74, 87, 91, 110, 121, 123, 138, 152, 166, 187, 202, 208, 218, 223, 234, 276, 288, 315, 320, 365, 374, 394, 411, 432, 455, 480, 492, 541, 547, 567, 574, 626, 685, 708, 716, 732, 764, 772, 818, 851

Offset: 1

Jesse H. Crotts, Aug 28 2016

nonn

Both p and p+2 must appear in the indicated range, and a prime can only be used once (so (3, 5) and (5, 7) can't both be used).
It appears that there should be more twin primes between prime(n) and prime(n)^2 as n increases. Specifically this sequence should be strictly increasing.
Indeed even the number of twin primes between prime(n)^2 and prime(n+1)^2 (A057767) seems to have a lower bound of about n/11. - M. F. Hasler, Jun 27 2019

			For n=3, prime(3)=5 because it is the 5th prime. There are 3 twin prime subsets on the set {5,6,7,...,24,25} so the 3rd term is 3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A001097, A079047, A143738.

Table[Function[w, Length@ Select[Prime[Range @@ w], Function[p, And[# - p == 2, # < Prime@ Last@ w] &@ NextPrime@ p]]]@ {n, PrimePi[Prime[n]^2]}, {n, 55}] (* Michael De Vlieger, Aug 30 2016 *)
ntp[n_]:=Count[Partition[Select[Range[Prime[n],Prime[n]^2],PrimeQ],2,1], ?(#[[2]]-#[[1]]==2&)]; Join[{0,1},Array[ntp,60,3]] (* _Harvey P. Dale, Nov 01 2016 *)

a(n)=if(n<3,return(n-1)); my(p=prime(n),q=p,s); forprime(r=q+1,p^2, if(r-q==2, s++); q=r); s \\ Charles R Greathouse IV, Aug 28 2016

More terms from Charles R Greathouse IV, Aug 28 2016

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

cp's OEIS Frontend

A273257 Number of twin primes between prime(n) and prime(n)^2.

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