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A053816 Another version of the Kaprekar numbers (A006886): 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 and n an m-digit number. 6
1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4950, 5050, 7272, 7777, 9999, 17344, 22222, 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

Consider an m-digit number n. Square it and add the right m digits to the left m or m-1 digits. If the resultant sum is n, then n is a term of the sequence.

4879 and 5292 are in A006886 but not in this version.

Shape of plot (see links) seems to consist of line segments whose lengths along the x-axis depend on the number of unitary divisors of 10^m-1 which is equal to 2^w if m is a multiple of 3 or 2^(w+1) otherwise, where w is the number of distinct prime factors of the repunit of length m (A095370). w for m = 60 is 20, whereas w <= 15 for m < 60. This leads to the long segment corresponding to m = 60. - Chai Wah Wu, Jun 02 2016

REFERENCES

D. R. Kaprekar, On Kaprekar numbers, J. Rec. Math., 13 (1980-1981), 81-82.

D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 151.

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..50000

D. E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.

R. Munafo, Kaprekar Sequences

Eric Weisstein's World of Mathematics, Kaprekar Number

Chai Wah Wu, Semilog plot of A053816(n), n = 1..1003371 (all terms with m <= 60).

Chai Wah Wu, Plot of A053816(n), n = 1..1003371 (all terms with m <= 60).

Index entries for Colombian or self numbers and related sequences

EXAMPLE

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

MATHEMATICA

kapQ[n_]:=Module[{idn2=IntegerDigits[n^2], len}, len=Length[idn2]; FromDigits[ Take[idn2, Floor[len/2]]]+FromDigits[Take[idn2, -Ceiling[len/2]]]==n]; Select[Range[540000], kapQ] (* Harvey P. Dale, Aug 22 2011 *)

ktQ[n_] := ((x = n^2) - (z = FromDigits[Take[IntegerDigits[x], y = -IntegerLength[n]]]))*10^y + z == n; Select[Range[540000], ktQ] (* Jayanta Basu, Aug 04 2013 *)

Select[Range[540000], Total[FromDigits/@TakeDrop[IntegerDigits[#^2], Floor[ IntegerLength[ #^2]/2]]] ==#&] (* The program uses the TakeDrop function from Mathematica version 10 *) (* Harvey P. Dale, Jun 03 2016 *)

PROG

(Haskell)

a053816 n = a053816_list !! (n-1)

a053816_list = 1 : filter f [4..] where

   f x = length us - length vs <= 1 &&

         read (reverse us) + read (reverse vs) == x

         where (us, vs) = splitAt (length $ show x) (reverse $ show (x^2))

-- Reinhard Zumkeller, Oct 04 2014

CROSSREFS

Cf. A006886, A037042, A053394, A053395, A053396, A053397, A045913, A003052, A055642.

Sequence in context: A087969 A044111 A006886 * A044492 A207359 A243090

Adjacent sequences:  A053813 A053814 A053815 * A053817 A053818 A053819

KEYWORD

nonn,nice,base,easy

AUTHOR

Robert Munafo

EXTENSIONS

More terms from Michel ten Voorde (seqfan(AT)tenvoorde.org), Apr 11 2001

STATUS

approved

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Last modified December 3 18:47 EST 2016. Contains 278745 sequences.