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A283247 a(n) is the smallest prime number whose representation contains as a substring the first n digits of Pi in base 10.

%I A283247 #60 Mar 23 2025 18:26:07
%S A283247 3,31,13147,73141,314159,314159,131415923,1314159269,23141592653,
%T A283247 23141592653,314159265359,3141592653581,213141592653589,
%U A283247 1131415926535897,9314159265358979,173141592653589793,3141592653589793239,3141592653589793239,314159265358979323861
%N A283247 a(n) is the smallest prime number whose representation contains as a substring the first n digits of Pi in base 10.
%C A283247 Pi progresses as 3, 31, 314, 3141, hence minimal prime numbers that do this are 3, 31, 13147, 73141. While there are other primes that contain, say, 314, the prime number, 13147 is the first prime to do so.
%C A283247 It is probably provable that this is an infinite sequence. Notice that 314159 appears twice in the sequence since 314159 is the smallest prime that contains 31415 as well as 314159.
%C A283247 a(n) exists for all n since for sufficiently large k, the k-th prime gap < prime(k)^d for some d < 1, so for a fixed number a, the next prime after a*10^m will be less than (a+1)*10^m for sufficiently large m and thus contain a as a substring. - _Chai Wah Wu_, Feb 22 2018
%H A283247 Chai Wah Wu, <a href="/A283247/b283247.txt">Table of n, a(n) for n = 1..997</a>
%H A283247 Manan Shah, <a href="http://mathmisery.com/wp/2017/07/19/summer-excursion-3-a-pi-containing-prime-number-sequence-challenge/">A Pi Containing Prime Number Sequence</a>
%e A283247 a(4) = 73141 since 73141 is the smallest prime number that contains 3141 (the first 4 digits of Pi).
%e A283247 a(5) = 314159 since 314159 is the smallest prime number that contains 31415.
%e A283247 a(6) = 314159 since 314159 is the smallest prime number that contains 314159.
%t A283247 pp[n_] := If[PrimeQ@n, n, Block[{d = IntegerDigits@n, p, s, t}, p = 10^Length[d]; s = Select[Join[Range[9] p + n, {1,3,7,9} + 10 n], PrimeQ]; If[s != {}, Min@s, s = NextPrime[100 n]; t = Join[If[Floor[s/100] == n, {s}, {}], Range[10, 99] p + n, FromDigits /@ Flatten /@ Tuples[{Range@9, {d}, {1, 3, 7, 9}}]]; s = Select[t, PrimeQ]; If[s == {}, 0, Min@s]]]]; Table[pp[Floor[10^e Pi]], {e, 0, 18}] (* _Giovanni Resta_, Jul 21 2017 *)
%o A283247 (Python)
%o A283247 pi_digits = pi_digit_generator #user-defined generator for producing next digit of Pi
%o A283247 next_digit = pi_digits.next() #first call, so next_digit = 3
%o A283247 primes = prime_generator #user-defined generator for producing next prime
%o A283247 current_prime = primes.next() #first call, so current_prime = 2
%o A283247 pi_progress = 0
%o A283247 while True:
%o A283247     pi_progress = pi_progress*10 + next_digit
%o A283247     while str(pi_progress) not in str(current_prime):
%o A283247         current_prime = primes.next()
%o A283247     print(pi_progress,current_prime)
%Y A283247 Cf. A005042, A282973.
%K A283247 nonn,base
%O A283247 1,1
%A A283247 _Manan Shah_, Jul 20 2017
%E A283247 a(7)-a(19) from _Giovanni Resta_, Jul 21 2017