A135945 - OEIS

%I #25 Feb 13 2023 18:01:11

%S 21,80,1972,26108,473297,7854396,144622191,2620370968

%N Number of primes with 2n digits made of adjacent primes with 2 digits.

%C Number of primes of the form: .. dd3dd2dd1dd0 = dd0 * 10^0 + dd1 * 10^2 + dd2 * 10^4 + dd3 * 10^6 + ..., where dd0, dd1, dd2, dd3, ... are primes with two digits. The i-th element of the sequence is the number of primes with i * 2 digits.

%C Approximation for the sum of the sequence up to a(k) for large values of k: Sum_{i=1..k} a(i) = 10^(2*k) / (2*k*log(10)) * (21/100)^(k-1).

%e a(2) = 80 because there are 80 numbers of the form dd1dd0 with dd0, dd1 prime numbers of 2 digits, i.e.: 1117, 1123, 1129, 1153, 1171, 1319, 1361, 1367, ..., 9767.

%o (Python)

%o from itertools import product

%o from sympy import isprime, primerange

%o def a(n):

%o     p2 = list(map(str, primerange(10, 100)))

%o     return sum(1 for p in product(p2, repeat=n) if isprime(int("".join(p))))

%o print([a(n) for n in range(1, 6)]) # _Michael S. Branicky_, Feb 13 2023

%Y Cf. A135944, A135946.

%K nonn,base

%O 1,1

%A _Giorgio Balzarotti_ & _Paolo P. Lava_, Dec 07 2007

%E a(7) from _Chai Wah Wu_, Nov 29 2015

%E a(8) from _Michael S. Branicky_, Feb 12 2023