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

Author

Robert Munafo

Keywords

• nonn
• nice
• base
• easy

Comments

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

Examples

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

References

• 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.

Links

• 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

Crossrefs

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.

Programs

Haskell

-- 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

Mathematica

(* 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 *)

PARI

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

Python

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

Formula

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

Extensions

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