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Reduced numbers with multiplicative persistence 11 in base 14.
+10
4
18693488093783, 152662885954879, 152949075509508, 305363914228295, 322719194103095, 489085121468831, 549494859818967, 2075869423190275, 2084591547337851, 2084591547385087, 2136925978841723, 2141285475971935, 2197979373249743, 2198602382114711, 2935000346492647
COMMENTS
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 conjectured to be finite. Each class representative number represents an infinite set of numbers with the same multiplicative persistence.If there exists more reduced numbers with multiplicative persistence 11, it will be larger than 14^100.
EXAMPLE
The number 18693488093783 represented in base 14, with A..D for 10..13 is 288AABCCCDDD. Other numbers with the same reduced number are for instance 2248AABCCCDDD, 28825AABCCCDDD, 288AABCC26DDD, 1288AABCCCDDD; or any number obtained by permutation of the digits of those numbers.
Smallest number with multiplicative persistence n in base 14.
+10
3
0, 14, 35, 54, 81, 135, 667, 2532, 130883, 499407, 397912927, 18693488093783, 82092087348200531993, 3393205899117928970481629894345
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.
MATHEMATICA
With[{s = Array[Length@ FixedPointList[Times @@ IntegerDigits[#, 14] &, #] - 2 &, 14^5]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Nov 13 2018 *)
Reduced numbers with multiplicative persistence 10 in base 14.
+10
3
397912927, 5449646691, 7987762047, 8408836868, 8416865813, 14311910772, 95358760419, 95359835871, 119207613951, 119319977811, 178002991683, 181182645587, 268593608191, 780359955604, 781941158165, 802715724229, 985373000485, 986970420491, 1068017311927, 1068025296336
COMMENTS
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.
If there exists more reduced numbers with multiplicative persistence 10, it will be larger than 14^100.
EXAMPLE
The number 397912927 represented in base 14, with A..D for 10..13 is 3ABBDDDD. Other numbers with the same reduced number are for instance 325BBDDDD or 13ABBDDDD; or any number obtained by permutation of the digits of those numbers.
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