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A003001 Smallest number of multiplicative persistence n.
(Formerly M4687)
54
0, 10, 25, 39, 77, 679, 6788, 68889, 2677889, 26888999, 3778888999, 277777788888899 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Probably finite.

The persistence of a number (A031346) is the number of times you need to multiply the digits together before reaching a single digit.

From David A. Corneth, Sep 23 2016: (Start)

For n > 1, the digit 0 doesn't occur. Therefore the digit 1 doesn't occur and all terms have digits in nondecreasing order.

a(n) consists of at most one three and at most one two but not both. If they contain both, they could be replaced with a single digit 6 giving a lesser number. Two threes can be replaced with a 9. Similarily, there's at most one four and one six but not both. Two sixes can be replaced with 49. A four and a six can be replaced with a three and an eight. For n > 2, an even number and a five don't occur together.

Summarizing, a term a(n) for n > 2 consists of 7's, 8's and 9's with a prefix of one of the following sets of digits: {{}, {2}, {3}, {4}, {6}, {2,6}, {3,5}, {5, 5,...}} [Amended by Kohei Sakai, May 27 2017]

No more up to 10^200. (End)

From Benjamin Chaffin, Sep 29 2016: (Start)

Let p(n) be the product of the digits of n, and P(n) be the multiplicative persistence of n. Any p(n) > 1 must have only prime factors from one of the two sets {2,3,7} or {3,5,7}. The following are true of all p(n) < 10^20000:

The largest p(n) with P(p(n))=10 is 2^4 * 3^20 * 7^5. The only other such p(n) known is p(a(11))=2^19 * 3^4 * 7^6.

The largest p(n) with P(p(n))=9 is 2^33 * 3^3 (12 digits).

The largest p(n) with P(p(n))=8 is 2^9 * 3^5 * 7^8 (12 digits).

The largest p(n) with P(p(n))=7 is 2^24 * 3^18 (16 digits).

The largest p(n) with P(p(n))=6 is 2^24 * 3^6 * 7^6 (16 digits).

The largest p(n) with P(p(n))=5 is 2^35 * 3^2 * 7^6 (17 digits).

The largest p(n) with P(p(n))=4 is 2^59 * 3^5 * 7^2 (22 digits).

The largest p(n) with P(p(n))=3 is 2^4 * 3^17 * 7^38 (42 digits).

The largest p(n) with P(p(n))=2 is 2^25 * 3^227 * 7^28 (140 digits).

All p(n) between 10^140 and 10^20000 have a persistence of 1, meaning they contain a 0 digit. (End)

Benjamin Chaffin's comments imply that there are no more terms up to 10^20585. For every number N between 10^200 with 10^20585 with persistence greater than 1, the product of the digits of N is between 10^140 and 10^20000, and each of these products has a persistence of 1.  -  David Radcliffe, Mar 22 2019

From A.H.M. Smeets, Nov 16 2018: (Start)

Let p_10(n) be the product of the digits of n in base 10. We can define an equivalence relation DP_10 on n by n DP_10 m if and only if p_10(n) = p_10(m); the name 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_10 if and only if p_10(n) = p_10(m), m >= n; i.e., if it is the smallest number of that class; it is also called the reduced number.

For any multiplicative persistence, except 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.

For multiplicative persistence 2, only the set of class representative numbers which end in the digit zero is infinite. The table of numbers of class representative numbers of different multiplicative persistence (mp) is given by:

               final digit

mp  total      0   1   2   3   4   5   6   7   8   9

====================================================

0      10      1   1   1   1   1   1   1   1   1   1

1      10      1   1   1   1   1   1   1   1   1   1

2     inf    inf   0   4   0   1   1   5   0   7   0

3   12199  12161   0   8   0   3   3   8   0  16   0

4     408    342   0  14   0   5   4  19   0  24   0

5     151     88   0   9   0   1   3  37   0  13   0

6      41     24   0   1   0   0   0  14   0   2   0

7      13      9   0   0   0   0   0   4   0   0   0

8       8      7   0   0   0   0   0   1   0   0   0

9       5      5   0   0   0   0   0   0   0   0   0

10      2      2   0   0   0   0   0   0   0   0   0

11      2      2   0   0   0   0   0   0   0   0   0

It is observed from this that for the reduced numbers with multiplicative persistence 1, the primes 11, 13, 17 and 19, will not occur in any trajectory of another (larger) number; i.e., all numbers represented by the reduced numbers 11, 13, 17 and 19 have a prime factor of at least 11 (conjectured from the observations).

Example for numbers represented by the reduced number 19: 91 = 7*13, 133 = 7*19, 313 is prime, 331 is prime, 119 = 7*17, 191 is prime, 911 is prime, 1133 = 11*103, 1313 = 13*101, 1331 = 11^3, 3113 = 11*283, 3131 = 31*101 and 3311 = 7*11*43.

In fact all trajectories can be projected to a trajectory in one of the ten trees with reduced numbers with roots 0..9, and the numbers represented by the reduced number of each leaf have a prime factor of at least 11 (as conjectured from the observations).

Example of the trajectory of 277777788888899 (see A121111) in the tree of reduced numbers (the unreduced numbers are given between brackets): 277777788888899 -> 3778888999 (4996238671872) -> 26888999 (438939648) -> 2677889 (4478976) -> 68889 (338688) -> 6788 (27648) -> 2688 (2688) -> 678 (768) -> 69 (336) -> 45 (54) -> 10 (20) -> 0. (End)

REFERENCES

Alex Bellos, Here's Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math, Free Press, 2010, page 176.

M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, pp. 170, 186.

C. A. Pickover, Wonders of Numbers, "Persistence", Chapter 28, Oxford University Press NY 2001.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 66.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

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

de Faria, Edson, and Charles Tresser, On Sloane's persistence problem, arXiv preprint arXiv:1307.1188 [math.DS], 2013.

de Faria, Edson, and Charles Tresser, On Sloane's persistence problem, Experimental Math., 23 (No. 4, 2014), 363-382.

Mark R. Diamond, Multiplicative persistence base 10: some new null results, 2011.

Brady Haran and Matt Parker, What's special about 277777788888899?, Numberphile video, 2019.

S. Perez, R. Styer, Persistence: A Digit Problem

W. Schneider, The Persistence of a Number

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Eric Weisstein's World of Mathematics, Multiplicative Persistence

Wikipedia, Persistence of a number

Susan Worst, Multiplicative persistence of base four numbers [Scanned copy of manuscript and correspondence, May 1980]

EXAMPLE

77 -> 49 -> 36 -> 18 -> 8 has persistence 4.

MATHEMATICA

lst = {}; n = 0; Do[While[True, k = n; c = 0; While[k > 9, k = Times @@ IntegerDigits[k]; c++]; If[c == l, Break[]]; n++]; AppendTo[lst, n], {l, 0, 7}]; lst (* Arkadiusz Wesolowski, May 01 2012 *)

PROG

(PARI) vecprod(w)=prod(i=1, #w, w[i]);

persistence(x)={my(y=digits(x), c=0); while(#y>1, y=digits(vecprod(y)); c++); return(c)}

firstTermsA003001(U)={my(ans=vector(U), k=(U>1), z); while(k+1<=U, if(persistence(z)==k, ans[k++]=z); z++); return(ans)}

\\ Finds the first U terms (is slow); R. J. Cano, Sep 11 2016

CROSSREFS

Cf. A031346 (persistence), A133500 (powertrain), A133048 (powerback), A006050, A007954, A031286, A031347, A033908, A046511, A121105-A121111.

Sequence in context: A002600 A087473 A014120 * A198377 A038350 A003344

Adjacent sequences:  A002998 A002999 A003000 * A003002 A003003 A003004

KEYWORD

nonn,nice,base,more,hard

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 14 12:45 EDT 2019. Contains 328006 sequences. (Running on oeis4.)