Number of Kaprekar

In Mathematical, a number of Kaprekar is a Nombre which, in a bases given, when it is high squared, can be separate in a left part and a right part (nonnull) such that the sum gives the initial number.

Exemples

703 is a number of Kaprekar bases 10 of them because 703 ² = 494.209 and that 494 + 209 = 703.

4879 is a number of Kaprekar bases 10 of them because 4879 ² = 23.804.641 and 04641 + 238 = 4879

The numbers of Kaprekar were mainly studied by D.R. Kaprekar, mathematician Indian.

N - number of Kaprekar

Either N a natural entirety not no one, K is a N - number of Kaprekar in the base has if and only if there exist two natural entireties U unspecified and

0 < v < a^n

such as

k^2 = u.a^n + v

k = U + v

The list of the first N - numbers of Kaprekar in base 10 is the following one:

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

In an inventory which makes, in 1980, D.R. Kaprekar, it surprisingly forgets all the numbers of the form $10^n - 1$ as well as numbers 181819 and 818181. The error is rectified in 1981 per Mr. Charosh who develops a method of generation of great numbers of Kaprekar.

In 2000, Douglas Iannucci, in the Journal off integer sequence shows that N - numbers of basic Kaprekar 10 are in bijection with the unit dividing of $10^n-1$ and shows how to obtain them starting from the decomposition of $10^n-1$ in factors first. It shows moreover that, if K is a N - number of Kaprekar, he is the same of $10^n - k$

Exemple

for N = 2,

10^2-1

= 99 which divides in 2 ways using unit dividers distinct from 1

99 = 9 × 11. However 45 is the smallest adequate multiple of 9 with 1 modulo 11 and 45 is a 2-number of Kaprekar.

99 = 11 × 9. However 55 is the smallest multiple of 11 adequate to 1 modulo 9 and 55 is a 2-number of Kaprekar

One notices moreover than

55 + 45 = 10^2

for N = 3,

10^3-1

= 999 which divides in 2 ways using unit dividers distinct from 1

999 = 27 × 37. However 297 is the smallest multiple of 27 adequate to 1 modulo 37 and 297 is a 3-number of Kaprekar.

999 = 37 × 27 and 703 is the smallest multiple of 37 adequate to 1 modulo 27 and 703 is a 3-number of Kaprekar.

Lastly, one notices that

297 + 703 = 10^3

Iannucci shows in addition that the numbers of Kaprekar bases 2 of them are all the perfect numbers even

Number of natural Kaprekar

Certain articles impose squared numbers of Kaprekar a decomposition in two parts of quasi-equal sizes. In fact, a natural entirety K of N figures is known as of natural Kaprekar if its square breaks up into a right part of N figures and a left part of N or N - 1 figures such as their nap gives K. By imposing this additional condition, the list of the numbers of Kaprekar is being reduced

List natural numbers of Kaprekar bases 10 of them:

1,9,45,55,99,297,703,999,2223,2728,4950,5050,7272,7777,9999,17344,22222,77778,….

See too

Algorithm of Kaprekar

References

D.R. Kaprekar, One Kaprekar numbers , J. Rec. Maths., 13 (1980-1981), 81-82.
Mr. Charosh, Sum Applications off Casting Out 999… 'S , Newspaper off Recreational Mathematics 14,1981-82, pp. 111-118
Demonstration of Douglas Iannucci in Newspaper off Integer Sequences 3,2000, Article 00.1.2
natural Numbers of Kaprekar '' ''

External bond

Continuation of the numbers of Kaprekar in On-line Encyclopedia off Integer Sequences (A006886 Continuation)