Multiplicative digital root

The cross product of a natural number is - similar to the sum of the digits - the product of its digits values ​​. The decimal cross product of 5496, for example, 5.4.9.6 = 1080th Just as the sum of the digits and the cross product is dependent on the number system used. In each number system digit numbers correspond to their own cross product.

Graph History

The graph of the cross- product feature that every natural number n their cross product q (n) assigns, possesses a characteristic course. It consists of consecutive spikes that reach ever higher peaks. Between these spikes fall q (n) repeatedly to 0; namely whenever n in at least one digit is 0.

This behavior occurs in any of magnitude on - the range 0 ≤ n ≤ 10 also forms a wave as 0 ≤ n ≤ 10,000. In this way occurs in the graph of q (n) self-similarity. When considering a power of ten, the first two prongs are always the same size, the following eight represent the two -, three -, four times, etc. of the first gendarme dar.

The smallest function value q (n ) is 0, a limit does not exist.

Iterated cross product

The sequence generated is a series of numbers in which each number is the cross product of its predecessor, thus ending for each multi-digit starting number after finitely many steps at a single digit number. This is due to the fact that the cross- product of a multi-digit number is always less than the number itself

3784 → 3.7.8.4 = 672 6.7.2 = 84 → → → 8.4 = 32 3.2 = 6 75664 → 5040 → 5.0.4.0 7.5.6.6.4 = 0 =

The number of steps is called persistence (English multiplicative persistence ) of a number. Thus has the perseverance 3784 4 and 75664 perseverance 2 -digit number that you will receive at the end of the chain, as multiplicative digital root (Eng. " multiplicative numerals root " ) is called.

For the following Beharrlichkeiten each smallest start numbers known ( sequence A003001 in OEIS ) in the decimal system. A number with the perseverance of 12 is not yet known.