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A281424 Numbers k such that 16*(10^k - 1)/3 + 1 is prime.

%I A281424 #27 Apr 16 2024 04:02:43
%S A281424 6,23,65,82,108,188,300,342,401,584,1570,4119,10030,24870,34710
%N A281424 Numbers k such that 16*(10^k - 1)/3 + 1 is prime.
%C A281424 All prime numbers of the form 16*(10^k - 1)/3 + 1 are terms of A002476.
%C A281424 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 A281424 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 A281424 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 A281424 Equivalently defined as primes of the form (16*10^k-13)/3. - _Tyler Busby_, Apr 16 2024
%H A281424 Makoto Kamada, <a href="https://stdkmd.net/nrr/prime/prime_difficulty.txt">Search for 53w29</a>.
%e A281424 For k = a(1) = 6, 16*(10^6 - 1)/3 + 1 = 5333329 and 16*(10^6 - 1)/3 + 1 is prime.
%t A281424 Select[Range@ 3000, PrimeQ[16 (10^# - 1)/3 + 1] &] (* _Michael De Vlieger_, Jan 23 2017 *)
%o A281424 (Python)
%o A281424 from sympy import isprime
%o A281424 def afind(limit, startk=1):
%o A281424     pow10 = 10**startk
%o A281424     for k in range(startk, limit+1):
%o A281424         if isprime(16*(pow10 - 1)//3 + 1): print(k, end=", ")
%o A281424         pow10 *= 10
%o A281424 afind(600) # _Michael S. Branicky_, Aug 17 2021
%Y A281424 Cf. A002476.
%K A281424 nonn,more
%O A281424 1,1
%A A281424 _Sergey Pavlov_, Jan 21 2017
%E A281424 a(12) from _Michael S. Branicky_, Aug 17 2021
%E A281424 a(13)-a(14) from _Michael S. Branicky_, Apr 06 2023
%E A281424 a(15) from Kamada data by _Tyler Busby_, Apr 16 2024