OFFSET

1,1

COMMENTS

Let p_15(n) be the product of the digits of n in base 15. We can define an equivalence relation DP_15 on n by n DP_15 m if and only if p_15(n) = p_15(m); the naming DP_b for the equivalence relation stands for "digits product for representation in base b". A number n is called the class representative number of class n/DP_15 if and only if p_15(n) = p_15(m), m >= n; i.e., the smallest number of that class; it is also called the reduced number.

For any multiplicative persistence, except the multiplicative persistence 2, the set of class representative numbers with that multiplicative persistence is conjectured to be finite. Each class representative number represents an infinite set of numbers with the same multiplicative persistence.

If there exists a next reduced number m with multiplicative persistence 10, p(m) will be larger than 15^100, where p(m) is the product of the digits of m.

a(1) = A320721(10).

LINKS

A.H.M. Smeets, Table of n, a(n) for n = 1..13

EXAMPLE

The number 140410607143 represented in base 15, with A..E for 10..14 is 39BBCCCCCD. Other numbers with the same reduced number are for instance: 333BBCCCCCD, 39BB34CCCCD, 139BBCCCCCD; or any number obtained by permutation of the digits of those numbers.

CROSSREFS

KEYWORD

nonn,base

AUTHOR

A.H.M. Smeets, Oct 19 2018

STATUS

approved