A053397 -id:A053397 - cp's OEIS frontend

A006886 Kaprekar numbers: positive numbers n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1.

1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777, 9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170

Offset: 1

Robert Munafo

4879 and 5292 are in this sequence but not in A053816.
Digital root is either 1 or 9. - Ezhilarasu Velayutham, Jul 27 2019
Named after the Indian recreational mathematician Dattatreya Ramchandra Kaprekar (1905-1986). - Amiram Eldar, Jun 19 2021
The term a(11) = 4879 is the first not in subsequence A053816. - M. F. Hasler, Mar 28 2025

			703 is a Kaprekar number because 703 = 494 + 209, 703^2 = 494209.

D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., Vol. 13 (1980-1981), pp. 81-82.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.

Robert Gerbicz, Table of n, a(n) for n = 1..51514 [T. D. Noe computed terms 1 - 1019, Nov 10 2007; R. Gerbicz computed the first 51514 terms, Jul 28 2011]
Santanu Bandyopadhyay, Kaprekar Number, Indian Institute of Technology Bombay (Mumbai, India, 2020).
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 142.
Ömer Eğecioğlu and Bünyamin Şahin, On twin EP numbers, Transact. Comb. (2025) Vol. 14, Iss. 4, Art. No. 4, 261-270. See p. 262, also ResearchGate.
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
Hans Havermann, The first 11 million Kaprekar numbers (plus the region around the billionth).
Douglas E. Iannucci, The Kaprekar Numbers, Journal of Integer Sequences, Vol. 3 (2000), Article 1.2,
Douglas E. Iannucci and Bertrum Foster, Kaprekar Triples, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.8.
Robert Munafo, Kaprekar Sequences.
Rosetta Code, Kaprekar numbers.
Walter Schneider, Kaprekar Numbers, 2002.
Gérard Villemin's Almanach of Numbers, Nombres de Kaprekar
Eric Weisstein's World of Mathematics, Kaprekar Number.
Wikipedia, Kaprekar number.
Index entries for Colombian or self numbers and related sequences

See A053816 for another version.
Cf. A037042, A053394, A053395, A053396, A053397, A045913, A003052.
Cf. A193992 (where 10^n-1 occurs in A006886), A194232 (first differences).
Subsequence of A248353.

-- See A194218 for another version
a006886 n = a006886_list !! (n-1)
a006886_list = 1 : filter chi [4..] where
   chi n = read (reverse us) + read (reverse vs) == n where
       (us,vs) = splitAt (length $ show n) (reverse $ show (n^2))
-- Reinhard Zumkeller, Aug 18 2011

(* This Mathematica code computes five additional powers in order to be sure that all the Kaprekar numbers have been computed. This fix works for mx <= 50, which includes terms computed by Gerbicz. *)
Inv[a_, b_] := PowerMod[a, -1, b]; mx = 20; t = {1}; Do[h = 10^k - 1; d = Divisors[h]; d2 = Select[d, GCD[#, h/#] == 1 &]; If[Log[10, h] < mx, AppendTo[t, h]]; Do[q = d2[[i]]*Inv[d2[[i]], h/d2[[i]]]; If[Log[10, q] < mx, AppendTo[t, q]], {i, 2, Length[d2] - 1}], {k, mx + 5}]; t = Union[t] (* T. D. Noe, Aug 17 2011, Aug 18 2011 *)
kaprQ[\[Nu]_] := Module[{n = \[Nu]^2},
  MemberQ[Plus @@ # & /@
    Select[Table[{Floor[n/10^j], 10^j*FractionalPart[n/10^j]}, {j,
       IntegerLength@n - 1}], #[[2]] != 0 &], \[Nu]]];
Select[Range@1000000, kaprQ] (* Hans Rudolf Widmer, Oct 22 2021 *)

select( {is_A006886(n)=my(N=n^2,m=1);while(N>m*=10,n==N%m+N\m && m!=n && return(m));n==1}, [1..10^5]) \\ M. F. Hasler, Mar 28 2025

def is_A006886(n):
    m=1; return (N:=n**2)and any(n==sum(divmod(N,m:=m*10))!=m for _ in str(N))
print(upto_1e5 := [n for n in range(10**5)if is_A006886(n)]) # M. F. Hasler, Mar 28 2025

a(n) = A194218(n) + A194219(n) and A194218(n) concatenated with A194219(n) gives a(n)^2. - Reinhard Zumkeller, Aug 19 2011

More terms from Michel ten Voorde, Apr 11 2001
4879 and 5292 added by Larry Reeves (larryr(AT)acm.org), Apr 24 2001
38962 added by Larry Reeves (larryr(AT)acm.org), May 23 2002

A053816 Another version of the Kaprekar numbers (A006886): n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1 and n an m-digit number.

Original entry on oeis.org

1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 77778, 82656, 95121, 99999, 142857, 148149, 181819, 187110, 208495, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357

Offset: 1

Robert Munafo

Consider an m-digit number n. Square it and add the right m digits to the left m or m-1 digits. If the resultant sum is n, then n is a term of the sequence.
4879 and 5292 are in A006886 but not in this version.
Shape of plot (see links) seems to consist of line segments whose lengths along the x-axis depend on the number of unitary divisors of 10^m-1 which is equal to 2^w if m is a multiple of 3 or 2^(w+1) otherwise, where w is the number of distinct prime factors of the repunit of length m (A095370). w for m = 60 is 20, whereas w <= 15 for m < 60. This leads to the long segment corresponding to m = 60. - Chai Wah Wu, Jun 02 2016
If n*(n-1) is divisible by 10^m-1 then n is a term where m is the number of decimal digits in n. - Giorgos Kalogeropoulos, Mar 27 2025

			703 is Kaprekar because 703 = 494 + 209, 703^2 = 494209.

D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.

Chai Wah Wu, Table of n, a(n) for n = 1..50000
Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.
Douglas E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.
Robert Munafo, Kaprekar Sequences.
Eric Weisstein's World of Mathematics, Kaprekar Number.
Chai Wah Wu, Semilog plot of A053816(n), n = 1..1003371 (all terms with m <= 60).
Chai Wah Wu, Plot of A053816(n), n = 1..1003371 (all terms with m <= 60).
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A037042, A053394, A053395, A053396, A053397, A045913, A003052, A055642, A095370.

Fixed points of A380585.

a053816 n = a053816_list !! (n-1)
a053816_list = 1 : filter f [4..] where
   f x = length us - length vs <= 1 &&
         read (reverse us) + read (reverse vs) == x
         where (us, vs) = splitAt (length $ show x) (reverse $ show (x^2))
-- Reinhard Zumkeller, Oct 04 2014

kapQ[n_]:=Module[{idn2=IntegerDigits[n^2],len},len=Length[idn2];FromDigits[ Take[idn2,Floor[len/2]]]+FromDigits[Take[idn2, -Ceiling[len/2]]]==n]; Select[Range[540000],kapQ] (* Harvey P. Dale, Aug 22 2011 *)
ktQ[n_] := ((x = n^2) - (z = FromDigits[Take[IntegerDigits[x], y = -IntegerLength[n]]]))*10^y + z == n; Select[Range[540000], ktQ] (* Jayanta Basu, Aug 04 2013 *)
Select[Range[540000],Total[FromDigits/@TakeDrop[IntegerDigits[#^2], Floor[ IntegerLength[ #^2]/2]]] ==#&] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Jun 03 2016 *)
maxDigits=6; Flatten[Table[lst={};sub=Subsets@FactorInteger[v=10^d-1]; Do[a=Times@@Power@@@s; n=ChineseRemainder[{0,1},{a,v/a},1]; If[10^(d-1)<=n<10^d,AppendTo[lst,n]],{s,sub}];Union@lst,{d,maxDigits}]] (* Giorgos Kalogeropoulos, Mar 27 2025 *)

isok(n) = n == vecsum(divrem(n^2, 10^(1+logint(n, 10)))); \\ Ruud H.G. van Tol, Jun 02 2024

def is_A053816(n): return n==sum(divmod(n**2,10**len(str(n)))) and n
print(upto_1e5:=list(filter(is_A053816, range(10**5)))) # M. F. Hasler, Mar 28 2025

More terms from Michel ten Voorde, Apr 11 2001

A045913 Kaprekar numbers: numbers k such that k = q + r and k^2 = q*10^m + r, for some m >= 1, q >= 0 and 0 <= r < 10^m. Here q and r must both have the same number of digits.

Original entry on oeis.org

1, 9, 45, 55, 703, 4950, 5050, 7272, 7777, 77778, 82656, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357, 648648, 670033, 681318, 791505, 812890, 818181, 851851, 857143, 4444444, 4927941, 5072059, 5555556, 11111112, 36363636, 38883889, 44363341, 44525548, 49995000, 50005000

Offset: 1

N. J. A. Sloane

A variant of Kaprekar's original definition (A006886).

			703 is Kaprekar because 703 = 494 + 209, 703^2 = 494209.
11111112^2 = 123456809876544 = (1234568 + 9876544)^2. The two "halves" of the square have the same length here, although it's not m but rather m - 1.

D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 151.

Jinyuan Wang, Table of n, a(n) for n = 1..30047
D. E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.
Rosetta Code, Kaprekar numbers
Eric Weisstein's World of Mathematics, Kaprekar Number
Wikipedia, Kaprekar number
Index entries for Colombian or self numbers and related sequences

Cf. A006886, A037042, A053394, A053395, A053396, A053397, A003052, A248353.

More terms from Michel ten Voorde, Apr 13 2001
Definition clarified by Reinhard Zumkeller, Oct 05 2014
Definition modified and terms corrected by Max Alekseyev, Aug 06 2017

cp's OEIS Frontend

A006886 Kaprekar numbers: positive numbers n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1.

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A053816 Another version of the Kaprekar numbers (A006886): n such that n = q+r and n^2 = q*10^m+r, for some m >= 1, q >= 0 and 0 <= r < 10^m, with n != 10^a, a >= 1 and n an m-digit number.

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Programs

Haskell

Mathematica

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Extensions

A045913 Kaprekar numbers: numbers k such that k = q + r and k^2 = q*10^m + r, for some m >= 1, q >= 0 and 0 <= r < 10^m. Here q and r must both have the same number of digits.

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A053394 The full list of 3-Kaprekar numbers.

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A053395 Complete list of 4-Kaprekar numbers.

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A053396 The full list of 5-Kaprekar numbers.

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A333133 7-Kaprekar numbers.

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