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A006886
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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)
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26
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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
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OFFSET
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1,2
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COMMENTS
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4879 and 5292 are in this sequence but not in A053816.
Named after the Indian recreational mathematician Dattatreya Ramchandra Kaprekar (1905-1986). - Amiram Eldar, Jun 19 2021
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REFERENCES
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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.
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LINKS
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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.
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FORMULA
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EXAMPLE
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703 is a Kaprekar number because 703 = 494 + 209, 703^2 = 494209.
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MATHEMATICA
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(* 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]]];
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PROG
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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))
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CROSSREFS
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KEYWORD
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nonn,nice,base,easy
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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