This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A145875 #54 Apr 04 2025 22:35:14
%S A145875 1,9,55,99,999,7777,9999,22222,99999,999999,4444444,9999999,88888888,
%T A145875 99999999,999999999,1111111111,9999999999,55555555555,99999999999,
%U A145875 999999999999,7777777777777,9999999999999,22222222222222,99999999999999,999999999999999,4444444444444444,9999999999999999,88888888888888888,99999999999999999
%N A145875 Repdigit Kaprekar numbers.
%C A145875 Kaprekar numbers (A006886) all of whose digits are equal. - _N. J. A. Sloane_, Mar 26 2025
%C A145875 The only numbers where the repeated digit and the number of digits are the same are 1, 88888888 and 999999999.
%C A145875 Conjectures from _Daniel Mondot_, Mar 28 2025: (Start)
%C A145875 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.
%C A145875 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)
%C A145875 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
%H A145875 Daniel Mondot, <a href="/A145875/b145875.txt">Table of n, a(n) for n = 1..1000</a>
%H A145875 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_9">On Some Marvellous Numbers of Kaprekar</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315. See Sections 9.2.2 and 9.2.3.
%H A145875 Robert P. Munafo, <a href="http://www.mrob.com/pub/math/seq-kaprekar.html">Kaprekar sequences at MROB</a>
%F A145875 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
%F A145875 Length (= number of digits) of the n-th term is floor((n+2)*3/5). - _M. F. Hasler_, Mar 31 2025
%o A145875 (PARI) is_A145875(n)=is_A006886(n) && #Set(digits(n))==1 \\ _M. F. Hasler_, Mar 31 2025
%o A145875 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
%o A145875 (PARI) 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;
%o A145875 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
%Y A145875 Intersection of A006886 (Kaprekar numbers) and A010785 (repdigits).
%Y A145875 A382161 is a subsequence.
%Y A145875 A subsequence of A382163 (palindromic Kaprekar numbers).
%Y A145875 Cf. A002275 (repunits), A210434 (#digits(4^n), equals #digits(a(n+1)) for n < 98 but not beyond, due to log10(4) ~ 0.6).
%K A145875 nonn,base
%O A145875 1,2
%A A145875 Howard Berman (howard_berman(AT)hotmail.com), Oct 22 2008
%E A145875 More terms from Gupta (2025) added by _N. J. A. Sloane_, Mar 26 2025