cp's OEIS Frontend

A047658 Numbers k such that the initial k digits in decimal portion of Pi form a prime number.

%I A047658 #65 Mar 26 2025 11:29:52
%S A047658 5,12,281,547,6205,16350
%N A047658 Numbers k such that the initial k digits in decimal portion of Pi form a prime number.
%C A047658 Conjecture: this sequence is finite. - _Carlos Rivera_
%C A047658 Rivera's conjecture that this sequence is finite conflicts with heuristics; the next entry is almost certainly 6205, since floor((Pi-3)*10^6205) is (very) probably prime, though its proof may take decades. - _David Broadhurst_, Nov 08 2000
%C A047658 Floor((Pi-3)*10^6205) is a strong pseudoprime to all (1229) prime bases a < 10000 (the test took 45 minutes). - _Joerg Arndt_, Jan 16 2011
%C A047658 Terms for n>=5 are only probable primes. - _Dmitry Kamenetsky_, Aug 03 2015
%C A047658 Floor((Pi-3)*10^16350) is a probable prime, checked with 25 iterations of the Miller-Rabin test. - _Dmitry Kamenetsky_, Aug 05 2015
%C A047658 The next term is greater than 65400. - _Dmitry Kamenetsky_, Aug 09 2015
%C A047658 The next term is greater than 100000. - _Michael S. Branicky_, Sep 29 2024
%H A047658 Chris K. Caldwell, <a href="https://t5k.org/curios/page.php?number_id=551">Prime Curios: 14159...07021 (547-digits)</a>
%H A047658 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_19">Mystery of pi</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 19, 473-497.
%e A047658 5 gives 14159 (prime); 12 gives 141592653589 (prime) and so on.
%t A047658 nn=1000; d=RealDigits[Pi-3, 10, nn][[1]]; Select[Range[nn], PrimeQ[FromDigits[Take[d, #]]] &]
%o A047658 (PARI) is(n)=isprime((Pi-3)*10^n\1) \\ _Charles R Greathouse IV_, Aug 28 2015
%o A047658 (Python)
%o A047658 from sympy import S, isprime
%o A047658 pi_digits = str(S.Pi.n(10**5))[2:-1]
%o A047658 def afind():
%o A047658     kint = 0
%o A047658     for k in range(len(pi_digits)):
%o A047658         kint = 10*kint + int(pi_digits[k])
%o A047658         if isprime(kint):
%o A047658             print(k+1, end=", ")
%o A047658 afind() # _Michael S. Branicky_, Jan 29 2023
%Y A047658 Cf. A000796 (Pi), A060421, A064118.
%K A047658 hard,nice,nonn,base,more
%O A047658 1,1
%A A047658 _Carlos Rivera_