A382161 "Repunit" Kaprekar numbers.

1, 1111111111, 1111111111111111111, 1111111111111111111111111111, 1111111111111111111111111111111111111, 1111111111111111111111111111111111111111111111, 1111111111111111111111111111111111111111111111111111111, 1111111111111111111111111111111111111111111111111111111111111111

Offset

1,2

Comments

Kaprekar numbers (A006886) all of whose digits are 1's.

Not a very interesting sequence in itself, but needed for clarifying A145875.

[The data is copied from Gupta (2025); it would be nice to have it confirmed]

Links

Paolo Xausa, Table of n, a(n) for n = 1..100

Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315. See Section 9.2.2.

Index entries for linear recurrences with constant coefficients, signature (1000000001,-1000000000).

Formula

From Chai Wah Wu, May 31 2026: (Start)

a(n) = (10^(9*n-8)-1)/9.

a(n) = (10^9+1)*a(n-1) - 10^9*a(n-2) for n > 2.

G.f.: x*((10^9-10)*x/9 + 1)/((x - 1)*(10^9*x - 1)). (End)

Mathematica

(10^(9*Range[10] - 8) - 1)/9 (* Paolo Xausa, Jun 23 2026 *)

Prog

(Python)
def A382161(n): return (10**(9*n-8)-1)//9 # Chai Wah Wu, May 31 2026
(PARI) a(n)=(10^(9*n-8)-1)/9 \\ Charles R Greathouse IV, Jun 08 2026

Crossrefs

Cf. A006886, A145875.

Sequence in context: A272029 A095425 A038452 * A261510 A034650 A035525

Adjacent sequences: A382158 A382159 A382160 * A382162 A382163 A382164

Keyword

nonn,base,easy,changed

Author

N. J. A. Sloane, Mar 25 2025

Extensions

More terms from Chai Wah Wu, May 31 2026

Status

approved