A321135 Smallest number with multiplicative persistence n in base 14.

0, 14, 35, 54, 81, 135, 667, 2532, 130883, 499407, 397912927, 18693488093783, 82092087348200531993, 3393205899117928970481629894345

Offset: 0

Author

A.H.M. Smeets, Oct 28 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_14(n) be the product of the digits of n in base 14. We can define an equivalence relation DP_14 on n by n DP_14 m if and only if p_14(n) = p_14(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_14 if and only if p_14(n) = p_14(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 only reduced numbers with multiplicative persistence 13 in base 14 are 3393205899117928970481629894345, 322945973161112953421997463350995, 2267488046844813841159699333648004255, 2971191233814076922023513127030927045 and 333864146237706918678545467170069526940645387357 (in base 14, with A..D for 10..13: 55599999999999999AAAABBBBBB, 2888AAAAAAABBBBBCCCCCCCCCCCDD, 69999999999BBBBBBBBBBBBBBBBBDDDD, 8AAAAAAAABBBBBBBBBBCCCCCCCCCCCCD and 359999AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAABB).

The only reduced numbers with multiplicative persistence 12 in base 14 are given in A321136.

The only reduced numbers with multiplicative persistence 11 in base 14 are given in A321137.

The only reduced numbers with multiplicative persistence 10 in base 14 are given in A321138.

If there exists a number with multiplicative persistence 14, it will be larger than 14^100.

a(10) = A321138(1), a(11) = A321137(1) and a(12) = A321136(1).

Crossrefs

Cf. A003001 (base 10), A125582 (base 12), A320721 (base 15), A132161 (base 16), A321136, A321137, A321138.

Programs

Mathematica

With[{s = Array[Length@ FixedPointList[Times @@ IntegerDigits[#, 14] &, #] - 2 &, 14^5]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Nov 13 2018 *)