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
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
Programs
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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
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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
Comments