A382163 -id:A382163 - cp's OEIS frontend

A145875 Repdigit Kaprekar numbers.

1, 9, 55, 99, 999, 7777, 9999, 22222, 99999, 999999, 4444444, 9999999, 88888888, 99999999, 999999999, 1111111111, 9999999999, 55555555555, 99999999999, 999999999999, 7777777777777, 9999999999999, 22222222222222, 99999999999999, 999999999999999, 4444444444444444, 9999999999999999, 88888888888888888, 99999999999999999

Offset: 1

Howard Berman (howard_berman(AT)hotmail.com), Oct 22 2008

Kaprekar numbers (A006886) all of whose digits are equal. - N. J. A. Sloane, Mar 26 2025
The only numbers where the repeated digit and the number of digits are the same are 1, 88888888 and 999999999.
Conjectures from Daniel Mondot, Mar 28 2025: (Start)
The sequence a(n)%10 (i.e. the last (or any) digit of a(n)), is 15-periodic. the sequence would be : 1,9,5,9,9,7,9,2,9,9,4,9,8,9,9, repeating.
For n>15, a(n) can be constructed from a(n-15) by concatenating to it 9 times a digit of a(n-15). (End)
The above two conjectures are linked, they are easily proved using modular arithmetic, and correspond to the explicit formula given below. - M. F. Hasler, Mar 28 2025

Daniel Mondot, Table of n, a(n) for n = 1..1000
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315. See Sections 9.2.2 and 9.2.3.
Robert P. Munafo, Kaprekar sequences at MROB

Intersection of A006886 (Kaprekar numbers) and A010785 (repdigits).
A382161 is a subsequence.
A subsequence of A382163 (palindromic Kaprekar numbers).
Cf. A002275 (repunits), A210434 (#digits(4^n), equals #digits(a(n+1)) for n < 98 but not beyond, due to log10(4) ~ 0.6).

is_A145875(n)=is_A006886(n) && #Set(digits(n))==1 \\ M. F. Hasler, Mar 31 2025
apply( {A145875(n)=10^(n--*3\5+1)\9*if(bittest(5, n%5),[1,5,7,2,4,8][n%15*2\/5+1],9)}, [1..29]) \\ M. F. Hasler, Mar 28 2025

isk(k) = my(d=digits(k^2), nb=#d); if (nb%2, d=concat(0, d); nb++); fromdigits(Vec(d, nb/2)) + fromdigits(vector(nb/2, i, d[nb/2+i])) == k;
lista(nn) = my(list=List()); for (i=1, nn, for (d=1, 9, my(x = fromdigits(vector(i, k, d))); if (isk(x), listput(list, x)););); Vec(list); \\ Michel Marcus, Mar 29 2025

a(n) = d(n) * R(floor(n*3/5+2/5)), where R(n) = (10^n-1)/9 = A002275(n) and d = (1, 9, 5, 9, 9; 7, 9, 2, 9, 9; 4, 9, 8, 9, 9) repeating, where ";" is used just to emphasize the 3 similar subgroups of length 5, with 2nd, 4th and 5th element equal to 9. - M. F. Hasler, Mar 28 2025

Length (= number of digits) of the n-th term is floor((n+2)*3/5). - M. F. Hasler, Mar 31 2025

More terms from Gupta (2025) added by N. J. A. Sloane, Mar 26 2025

A382164 Palindromic Kaprekar numbers that are not repdigit Kaprekar numbers.

Original entry on oeis.org

909090909, 9090909090909090909090909090909, 81188118811881188118811881188118, 545545545545545545545545545545545, 277227722772277227722772277227722772, 505050505050505050505050505050505050505, 4040404040404040404040404040404040404040404040404

Offset: 1

N. J. A. Sloane, Mar 26 2025

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

Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315. See Section 9.2.4.

Cf. A006886.

A382163 is the union of this sequence and A145875.

cp's OEIS Frontend

A145875 Repdigit Kaprekar numbers.

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