%I #27 Apr 16 2024 04:02:43
%S 6,23,65,82,108,188,300,342,401,584,1570,4119,10030,24870,34710
%N Numbers k such that 16*(10^k - 1)/3 + 1 is prime.
%C All prime numbers of the form 16*(10^k - 1)/3 + 1 are terms of A002476.
%C For any k = a(n), the m-index of 16*(10^k - 1)/3 + 1 in sequence 6m+1 contains exactly a(n) digits, and each digit is 8. E.g., while k = a(1) = 6, 16*(10^6 - 1)/3 + 1 = 6*888888 + 1 = 5333329.
%C In any number of form 16*(10^k - 1)/3 + 1, its first digit is 5, its two last digits are 29, and each other digit that is between (5...29) is 3.
%C For k=1, k=2, k=3, the numbers of form 16*(10^k - 1)/3 + 1 are squares of the primes 7, 23, and 73, respectively (see A001248).
%C Equivalently defined as primes of the form (16*10^k-13)/3. - _Tyler Busby_, Apr 16 2024
%H Makoto Kamada, <a href="https://stdkmd.net/nrr/prime/prime_difficulty.txt">Search for 53w29</a>.
%e For k = a(1) = 6, 16*(10^6 - 1)/3 + 1 = 5333329 and 16*(10^6 - 1)/3 + 1 is prime.
%t Select[Range@ 3000, PrimeQ[16 (10^# - 1)/3 + 1] &] (* _Michael De Vlieger_, Jan 23 2017 *)
%o (Python)
%o from sympy import isprime
%o def afind(limit, startk=1):
%o pow10 = 10**startk
%o for k in range(startk, limit+1):
%o if isprime(16*(pow10 - 1)//3 + 1): print(k, end=", ")
%o pow10 *= 10
%o afind(600) # _Michael S. Branicky_, Aug 17 2021
%Y Cf. A002476.
%K nonn,more
%O 1,1
%A _Sergey Pavlov_, Jan 21 2017
%E a(12) from _Michael S. Branicky_, Aug 17 2021
%E a(13)-a(14) from _Michael S. Branicky_, Apr 06 2023
%E a(15) from Kamada data by _Tyler Busby_, Apr 16 2024