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 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. (Formerly M4625) 26
 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. 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 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). 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. 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 a Kaprekar number because 703 = 494 + 209, 703^2 = 494209. 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}], #[] != 0 &], \[Nu]]]; Select[Range@1000000, kaprQ] (* Hans Rudolf Widmer, Oct 22 2021 *) PROG (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 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. Sequence in context: A124983 A087969 A044111 * A053816 A290449 A045913 Adjacent sequences:  A006883 A006884 A006885 * A006887 A006888 A006889 KEYWORD nonn,nice,base,easy AUTHOR 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 STATUS approved

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Last modified September 26 21:54 EDT 2022. Contains 357050 sequences. (Running on oeis4.)