A093683 Number of pairs of twin primes <= 10^n-th prime.

4, 25, 174, 1270, 10250, 86027, 738597, 6497407, 58047180, 524733511, 4789919653, 44073509102, 408231310520

Offset

1,1

Comments

This sequence is >= the values of pi(10^n): 4, 25, 168, 1229, ... in A006880.

a(0) = 0. - Eduard Roure Perdices, Dec 23 2022

References

Enoch Haga, "Wandering through a prime number desert," Table 6, in Exploring prime numbers on your PC and the Internet, 2001 (ISBN 1-885794-17-7).

Links

Table of n, a(n) for n=1..13.

Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, The Bateman-Horn Conjecture: Heuristics, History, and Applications, arXiv:1807.08899 [math.NT], 2018-2019. See Table 5 p. 40.

Thomas R. Nicely, Twin prime count.

Index entries for sequences related to numbers of primes in various ranges

Formula

Count twin primes <= p_{10^n}: 10th prime, 100th prime, etc.

Example

a(1) = 4 because there are 4 twin primes <= 29, the 10th prime: (3,5), (5,7), (11,13), and (17,19). (29,31) is not counted because it is not entirely <= 29.

Mathematica

NextPrim[n_] := Block[{k = n + 1}, While[ !PrimeQ[k], k++ ]; k]; c = 0; p = q = 1; Do[l = Prime[10^n]; While[q <= l, If[p + 2 == q, c++ ]; p = q; q = NextPrim[p]]; Print[c], {n, 12}] (* Robert G. Wilson v, Apr 10 2004 *)

Prog

(Python)
from sympy import prime, sieve # use primerange for larger terms
def afind(terms):
  c, prevp = 0, 1
  for n in range(1, terms+1):
    for p in sieve.primerange(prevp+1, prime(10**n)+1):
      if prevp == p - 2: c += 1
      prevp = p
    print(c, end=", ")
afind(6) # Michael S. Branicky, Apr 25 2021

Crossrefs

See A049035 for another version. - R. J. Mathar, Sep 05 2008

Cf. A006880, A007508.

Sequence in context: A140177 A034494 A084210 * A006348 A213608 A369325

Adjacent sequences: A093680 A093681 A093682 * A093684 A093685 A093686

Keyword

nonn,more

Author

Enoch Haga, Apr 09 2004

Extensions

a(9) from Michael S. Branicky, Apr 25 2021

a(10) from Eduard Roure Perdices, May 08 2021

a(11) from Eduard Roure Perdices, Feb 03 2022

a(12) from Eduard Roure Perdices, Dec 23 2022

a(13) from Eduard Roure Perdices, Jan 24 2024

Status

approved