A045913 Kaprekar numbers: numbers k such that k = q + r and k^2 = q*10^m + r, for some m >= 1, q >= 0 and 0 <= r < 10^m. Here q and r must both have the same number of digits.

1, 9, 45, 55, 703, 4950, 5050, 7272, 7777, 77778, 82656, 318682, 329967, 351352, 356643, 390313, 461539, 466830, 499500, 500500, 533170, 538461, 609687, 643357, 648648, 670033, 681318, 791505, 812890, 818181, 851851, 857143, 4444444, 4927941, 5072059, 5555556, 11111112, 36363636, 38883889, 44363341, 44525548, 49995000, 50005000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy

Comments

A variant of Kaprekar's original definition (A006886).

Examples

			703 is Kaprekar because 703 = 494 + 209, 703^2 = 494209.
11111112^2 = 123456809876544 = (1234568 + 9876544)^2. The two "halves" of the square have the same length here, although it's not m but rather m - 1.
		

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, p. 151.

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..30047
• D. E. Iannucci, The Kaprekar numbers, J. Integer Sequences, Vol. 3, 2000, #1.2.
• Rosetta Code, Kaprekar numbers
• Eric Weisstein's World of Mathematics, Kaprekar Number
• Wikipedia, Kaprekar number
• Index entries for Colombian or self numbers and related sequences

Crossrefs

Cf. A006886, A037042, A053394, A053395, A053396, A053397, A003052, A248353.

Extensions

More terms from Michel ten Voorde, Apr 13 2001

Definition clarified by Reinhard Zumkeller, Oct 05 2014

Definition modified and terms corrected by Max Alekseyev, Aug 06 2017