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

Author

Enoch Haga, Apr 09 2004

Keywords

• nonn
• more

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

Examples

			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.
		

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

• 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

Crossrefs

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

Cf. A006880, A007508.

Programs

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 *)

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

Formula

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

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