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A040025 a(n) is the number of prime palindromes with 2n+1 digits.

%I A040025 #34 Dec 19 2024 06:20:02
%S A040025 4,15,93,668,5172,42042,353701,3036643,27045226,239093865,2158090933,
%T A040025 19742800564,180815391365
%N A040025 a(n) is the number of prime palindromes with 2n+1 digits.
%H A040025 Shyam Sunder Gupta, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;d9a588c5.0602">Palindromic Primes up to 10^19</a>.
%H A040025 Shyam Sunder Gupta, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;bdeb9ea2.0903">Palindromic Primes up to 10^21</a>.
%H A040025 Shyam Sunder Gupta, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;b79493a6.1310">Palindromic Primes up to 10^23</a>.
%H A040025 Shyam Sunder Gupta, <a href="https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;a27957b6.2412&amp;S=">Palindromic Primes up to 10^25</a>.
%e A040025 a(1)=15 because Number of prime palindromes with 3 digits is 15. [_Shyam Sunder Gupta_, Mar 14 2009]
%o A040025 (PARI) a(n) = {my(nb = 0); forprime(p=10^(2*n), 10^(2*n+1)-1, if (eval(concat(Vecrev(Str(p)))) == p, nb++);); nb;} \\ _Michel Marcus_, Jul 24 2015
%o A040025 (Python)
%o A040025 from sympy import isprime
%o A040025 from itertools import product
%o A040025 def candidate_pals(n): # of length 2n + 1
%o A040025   if n == 0: yield from [2, 3, 5, 7]; return # one-digit primes
%o A040025   for rightbutend in product("0123456789", repeat=n-1):
%o A040025     rightbutend = "".join(rightbutend)
%o A040025     for end in "1379": # multi-digit primes must end in 1, 3, 7, or 9
%o A040025       left = end + rightbutend[::-1]
%o A040025       for mid in "0123456789": yield int(left + mid + rightbutend + end)
%o A040025 def a(n): return sum(isprime(p) for p in candidate_pals(n))
%o A040025 print([a(n) for n in range(6)]) # _Michael S. Branicky_, Apr 15 2021
%Y A040025 Subsequence of A016115, which is the main entry.
%K A040025 nonn,hard,base,more
%O A040025 0,1
%A A040025 _Patrick De Geest_
%E A040025 a(9) from _Shyam Sunder Gupta_, Feb 12 2006
%E A040025 a(10) from _Shyam Sunder Gupta_, Mar 14 2009
%E A040025 a(11) from _Shyam Sunder Gupta_, Oct 05 2013
%E A040025 a(12) from _Shyam Sunder Gupta_, Dec 19 2024