A320721 Smallest number with multiplicative persistence n in base 15.

0, 15, 38, 58, 89, 582, 1964, 19526, 596667, 30104309, 140410607143, 3753516452901780134

Offset

0,2

Comments

Probably finite.

The persistence of a number is the number of times you need to multiply the digits together before reaching a single digit.

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 supposed to be finite. Each class representative number represents an infinite set of numbers with the same multiplicative persistence.

The known reduced numbers with multiplicative persistence 11 in base 15 are 3753516452901780134 and 166262836503982547199778 (in base 15, with A..E for 10..14: 88899BBBBDDDDDDE and 77777777777777BBBBBD).

The known reduced numbers with multiplicative persistence 10 in base 15 are given in A320722.

The known reduced numbers with multiplicative persistence 9 in base 15 are given in A320723.

If there exists a number m with multiplicative persistence 12, p(m) will be larger than 15^100.

a(9) = A320723(1) and a(10) = A320722(1).

Links

Table of n, a(n) for n=0..11.

Mathematica

With[{s = Array[Length@ FixedPointList[Times @@ IntegerDigits[#, 15] &, #] - 2 &, 15^5]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Nov 13 2018 *)

Crossrefs

Cf. A003001 (base 10), A125582 (base 12), A132161 (base 16), A320722, A320723.

Sequence in context: A321383 A039605 A032654 * A086096 A272189 A336560

Adjacent sequences: A320718 A320719 A320720 * A320722 A320723 A320724

Keyword

nonn,more,base

Author

A.H.M. Smeets, Oct 19 2018

Status

approved