A320721 Smallest number with multiplicative persistence n in base 15.

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

Offset: 0

Author

A.H.M. Smeets, Oct 19 2018

Keywords

• nonn
• more
• base

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).

Crossrefs

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

Programs

Mathematica

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