

A006886


Kaprekar 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.
(Formerly M4625)


20



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

4879 and 5292 are in this sequence but not in A053816.
The 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.  T. D. Noe, Aug 18 2011


REFERENCES

Douglas E. Iannucci and Bertrum Foster, Kaprekar Triples, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.8.
D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (19801981), 8182.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 151.


LINKS

T. D. Noe and 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]
Hans Havermann, The first 11 million Kaprekar numbers (plus the region around the billionth)
D. E. Iannucci, Journal of Integer Sequences, Vol. 3, 2000, #1.2, The Kaprekar Numbers
Douglas E. Iannucci and Bertrum Foster, Kaprekar Triples, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.8.
R. Munafo, Kaprekar Sequences
W. Schneider, Kaprekar Numbers
G. Villemin's Almanach of Numbers, Nombres de Kaprekar
Eric Weisstein's World of Mathematics, Kaprekar Number
Rosetta Code, Kaprekar numbers
Wikipedia, Kaprekar_number
Index entries for Colombian or self numbers and related sequences


FORMULA

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


EXAMPLE

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


MATHEMATICA

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 *)


PROG

(Haskell)
a006886 n = a006886_list !! (n1)
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
 See A194218 for another version


CROSSREFS

See A053816 for another version.
Cf. A037042, A053394, A053395, A053396, A053397, A045913, A003052.
Cf. A193992 (where 10^n1 occurs in A006886), A194232 (first differences).
Subsequence of A248353.
Sequence in context: A124983 A087969 A044111 * A053816 A044492 A207359
Adjacent sequences: A006883 A006884 A006885 * A006887 A006888 A006889


KEYWORD

nonn,nice,base,easy


AUTHOR

Robert Munafo


EXTENSIONS

More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org) 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


STATUS

approved



