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Number of pairs of twin primes whose smaller element is <= 10^n-th prime.
1

%I #42 Jan 24 2024 08:01:01

%S 5,25,174,1270,10250,86027,738597,6497407,58047180,524733511,

%T 4789919653,44073509102,408231310520

%N Number of pairs of twin primes whose smaller element is <= 10^n-th prime.

%C a(0) = 0. - _Eduard Roure Perdices_, Dec 23 2022

%H Soren Laing Aletheia-Zomlefer, Lenny Fukshansky, and Stephan Ramon Garcia, <a href="https://arxiv.org/abs/1807.08899">The Bateman-Horn Conjecture: Heuristics, History, and Applications</a>, arXiv:1807.08899 [math.NT], 2018-2019. See Table 5 p. 40.

%H <a href="/index/Pri#primepop">Index entries for sequences related to numbers of primes in various ranges</a>

%e a(1) = 5 since the 10th prime is 29 and the first 5 twin primes are {3,5}, {5,7}, {11,13}, {17,19} and {29,31}.

%o (Python)

%o from sympy import prime, sieve # use primerange for larger terms

%o def afind(terms):

%o c, prevp = 0, 1

%o for n in range(1, terms+1):

%o for p in sieve.primerange(prevp+1, prime(10**n)+3):

%o if prevp == p - 2: c += 1

%o prevp = p

%o print(c, end=", ")

%o afind(6) # _Michael S. Branicky_, Apr 25 2021

%Y See A093683 for another version.

%Y Cf. A001359, A006988.

%K nonn,more

%O 1,1

%A Dennis S. Kluk (mathemagician(AT)ameritech.net)

%E More terms from _Labos Elemer_, Oct 13 2000; _Jud McCranie_, Dec 30 2000

%E a(9) from _Michael S. Branicky_, Apr 25 2021

%E a(10) from _Eduard Roure Perdices_, May 08 2021

%E a(11) from _Eduard Roure Perdices_, Feb 03 2022

%E a(12) from _Eduard Roure Perdices_, Dec 23 2022

%E a(13) from _Eduard Roure Perdices_, Jan 24 2024