cp's OEIS Frontend

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.

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A053395 Complete list of 4-Kaprekar numbers.

Original entry on oeis.org

1, 2223, 2728, 4950, 5050, 7272, 7777, 9999
Offset: 1

Views

Author

N. J. A. Sloane, Jan 07 2000

Keywords

Links

Crossrefs

Cf. A006886, A037042, A053394, A053396, A053397, A333133.

A053396 The full list of 5-Kaprekar numbers.

Original entry on oeis.org

1, 4879, 17344, 22222, 77778, 82656, 95121, 99999
Offset: 1

Views

Author

N. J. A. Sloane, Jan 07 2000

Keywords

Links

Crossrefs

Cf. A006886, A037042, A053394, A053395, A053397, A333133.

A053397 The full list of 6-Kaprekar numbers.

Original entry on oeis.org

1, 5292, 38962, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357, 648648, 670033, 681318, 791505, 812890, 818181, 851851, 857143, 961038, 994708, 999999
Offset: 1

Views

Author

N. J. A. Sloane, Jan 07 2000

Keywords

Links

Crossrefs

Cf. A006886, A037042, A053394, A053395, A053396, A333133.

A118938 Sub-Kaprekar numbers (2): n such that n=r-q and n^2=q*10^m+r, for some m>=1, q>=0, 0<=r<10^m, with n not a power of 10.

Original entry on oeis.org

78, 287, 364, 1096, 18183, 336634, 2727274, 19138757, 23529412, 25974026, 97744361, 120879122, 140017878, 165991904, 237762239, 288553552, 307692308, 333666334, 405436669, 428571430, 440553516, 447710186, 454545455, 473684212
Offset: 1

Views

Author

Giovanni Resta, May 06 2006

Keywords

Examples

			287^2 = 82369 and 369-82 = 287.
A larger example is 1980198021^2 = 3921184202372316441, and 2372316441-392118420 = 1980198021.

Links

Crossrefs

Cf. A006886, A118936, A118937, A228381.

A248353 Kaprekar numbers, allowing powers of 10: n such that n=q+r and n^2=q*10^m+r, for some m >= 1, q>=0 and 0<=r<10^m.

Original entry on oeis.org

1, 9, 10, 45, 55, 99, 100, 297, 703, 999, 1000, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 10000, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 100000, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539
Offset: 1

Views

Author

Reinhard Zumkeller, Oct 05 2014

Keywords

Comments

Powers of 10 were excluded in Kaprekar's original definition (A006886), see also A045913.

Links

Crossrefs

Cf. A006886 (subsequence), A045913, A053816, A011557, A102766.

Programs

  • Haskell
    a248353 n = a248353_list !! (n-1)
a248353_list = filter k [1..] where
   k x = elem x $ map (uncurry (+)) $
         takeWhile ((> 0) . fst) $ map (divMod (x ^ 2)) a011557_list

Formula

a(n) = sqrt(A102766(n)).

A118936 Sub-Kaprekar numbers: k such that k = |q - r| and k^2 = q*10^m + r, for some m >= 1, q >= 0, 0 <= r < 10^m, with k not a power of 10.

Original entry on oeis.org

11, 78, 101, 287, 364, 1001, 1078, 1096, 1287, 1364, 10001, 11096, 18183, 100001, 118183, 336634, 1000001, 1336634, 2727274, 10000001, 12727274, 19138757, 23529412, 25974026, 97744361, 100000001, 120879122, 123529412, 140017878
Offset: 1

Views

Author

Giovanni Resta, May 06 2006; corrected May 12 2006

Keywords

Comments

Union of A118937 and A118938.

Examples

			287^2 = 82369 and |82 - 369| = 287, so 287 is a term.
1287^2 = 1656369 and |1656 - 369| = 1287, so 1287 is a term.

Crossrefs

Cf. A006886, A118937, A118938.

Programs

  • Mathematica
    f[n_] := !IntegerQ@Log[10,n] && Block[{p = 10^Range@Log[10,n^2]}, 0 == Times@@(n-Abs[Floor[n^2/p]-Mod[n^2,p]])]; Select[Range@400000,f]

A118937 Sub-Kaprekar numbers (1): n such that n=q-r and n^2=q*10^m+r, for some m>=1, q>=0, 0<=r<10^m, with n not a power of 10.

Original entry on oeis.org

11, 101, 1001, 1078, 1287, 1364, 10001, 11096, 100001, 118183, 1000001, 1336634, 10000001, 12727274, 100000001, 123529412, 1000000001, 1019138757, 1025974026, 1097744361, 1120879122, 1140017878, 1165991904, 1237762239, 1288553552
Offset: 1

Views

Author

Giovanni Resta, May 06 2006

Keywords

Examples

			1287^2 = 1656369 and 1656-369 = 1287.
A larger example: 1594563333^2 = 2542632222948068889 and
2542632222-948068889=1594563333.

Crossrefs

Cf. A006886, A118936, A118938.

A145875 Repdigit Kaprekar numbers.

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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

Links

Crossrefs

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).

Programs

  • PARI
    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
  • 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;
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

Formula

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

Extensions

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

A333133 7-Kaprekar numbers.

Original entry on oeis.org

1, 627615, 4444444, 4927941, 5072059, 5555556, 9372385, 9999999
Offset: 1

Views

Author

Eric Fox, Mar 09 2020

Keywords

Comments

No n-Kaprekar number k can have more than n digits because then the number to the left of the plus sign would have more digits than k itself, meaning the sum will always be greater than k.

Examples

			627615 is in this sequence because inserting a + before the 7th digit from the right of 627615^2 = 393900588225 yields 39390 + 0588225, which equals 627615 (the starting number).

Crossrefs

Cf. A006886, A037042, A053394, A053395, A053396, A053397.

A194218 Left part of the square of the n-th Kaprekar number.

Original entry on oeis.org

1, 8, 20, 30, 98, 88, 494, 998, 494, 744, 238, 2450, 2550, 28, 5288, 6048, 9998, 3008, 4938, 1518, 60494, 68320, 90480, 99998, 20408, 21948, 33058, 35010, 43470, 101558, 108878, 123448, 127194, 152344, 213018, 217930, 249500, 250500, 284270, 289940, 371718
Offset: 1

Views

Author

Reinhard Zumkeller, Aug 19 2011

Keywords

Comments

a(n) + A194219(n) = A006886(n) and
concatenation of a(n) and A194219(n) = A006886(n)^2.

Links

Programs

  • Haskell
    import Data.List (find)
import Data.Maybe (mapMaybe)
a194218 n = a194218_list !! (n-1)
a194218_list = map fst kaprekarPairs
a194219 n = a194219_list !! (n-1)
a194219_list = map snd kaprekarPairs
a006886 n = a006886_list !! (n-1)
a006886_list = map (uncurry (+)) kaprekarPairs
kaprekarPairs = (1,0) : (mapMaybe (\n -> kSplit n $ splits (n^2)) [1..])
   where kSplit x = find (\(left, right) -> left + right == x)
         splits q = no0 . map (divMod q) $ iterate (10 *) 10
         no0 = takeWhile ((> 0) . fst) . filter ((> 0) . snd)
-- Cf. Rosetta link.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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