A053816 - OEIS

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.

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, 538461, 609687, 643357

(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.

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

LINKS

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

Shyam Sunder Gupta, On Some Marvellous Numbers of Kaprekar, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 9, 275-315.

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

Robert 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

(PARI) isok(n) = n == vecsum(divrem(n^2, 10^(1+logint(n, 10)))); \\ Ruud H.G. van Tol, Jun 02 2024

CROSSREFS

Cf. A006886, A037042, A053394, A053395, A053396, A053397, A045913, A003052, A055642, A095370.

Sequence in context: A087969 A044111 A006886 * A290449 A045913 A044492