A350755 Numbers k such that 11...111 (with k 1's) = (10^k - 1)/9 is a Kaprekar prime.

19, 109297, 270343, 5794777

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

Table of n, a(n) for n=1..4.

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

Cf. A004022, A002275, A004023, A006886.

Sequence in context: A034207 A098970 A172751 * A013764 A078353 A347813

Adjacent sequences: A350752 A350753 A350754 * A350756 A350757 A350758

Keyword

nonn,hard,more

Author

Shyam Sunder Gupta, Jan 14 2022

Status

approved