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%I #20 Feb 06 2022 21:30:25
%S 19,109297,270343,5794777
%N Numbers k such that 11...111 (with k 1's) = (10^k - 1)/9 is a Kaprekar prime.
%C The terms that are repunit primes (including probable primes) are the only known Kaprekar primes, i.e., (10^19-1)/9 (prime), (10^109297-1)/9 (probable prime), (10^270343-1)/9 (probable prime), and (10^5794777-1)/9 (probable prime). Based on my investigations I conjecture that:
%C (i) A Kaprekar number can be prime only if it is a repunit prime with digital root 1.
%C (ii) Any repunit prime with digital root 1 is a Kaprekar prime [see Link: Puzzle 837. Kaprekar prime numbers].
%H Carlos Rivera, <a href="https://www.primepuzzles.net/puzzles/puzz_837.htm"> Puzzle 837. Kaprekar prime numbers</a>.
%e a(1) = 19 because for k=19, (10^k - 1)/9 = 1111111111111111111 (19-digit repunit prime) is the smallest Kaprekar prime as 1111111111111111111^2 = 1234567901234567900987654320987654321 and 123456790123456790 + 0987654320987654321 = 1111111111111111111.
%Y Cf. A004022, A002275, A004023, A006886.
%K nonn,hard,more
%O 1,1
%A _Shyam Sunder Gupta_, Jan 14 2022