OFFSET
1,1
COMMENTS
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:
(i) A Kaprekar number can be prime only if it is a repunit prime with digital root 1.
(ii) Any repunit prime with digital root 1 is a Kaprekar prime [see Link: Puzzle 837. Kaprekar prime numbers].
LINKS
Carlos Rivera, Puzzle 837. Kaprekar prime numbers.
EXAMPLE
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.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Shyam Sunder Gupta, Jan 14 2022
STATUS
approved