This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A003001 M4687 #193 Jul 06 2025 11:08:21
%S A003001 0,10,25,39,77,679,6788,68889,2677889,26888999,3778888999,
%T A003001 277777788888899
%N A003001 Smallest number of multiplicative persistence n.
%C A003001 Probably finite.
%C A003001 The persistence of a number (A031346) is the number of times you need to multiply the digits together before reaching a single digit.
%C A003001 From _David A. Corneth_, Sep 23 2016: (Start)
%C A003001 For n > 1, the digit 0 doesn't occur. Therefore the digit 1 doesn't occur and all terms have digits in nondecreasing order.
%C A003001 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.
%C A003001 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]
%C A003001 No more up to 10^200. (End)
%C A003001 From _Benjamin Chaffin_, Sep 29 2016: (Start)
%C A003001 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:
%C A003001 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.
%C A003001 The largest p(n) with P(p(n))=9 is 2^33 * 3^3 (12 digits).
%C A003001 The largest p(n) with P(p(n))=8 is 2^9 * 3^5 * 7^8 (12 digits).
%C A003001 The largest p(n) with P(p(n))=7 is 2^24 * 3^18 (16 digits).
%C A003001 The largest p(n) with P(p(n))=6 is 2^24 * 3^6 * 7^6 (16 digits).
%C A003001 The largest p(n) with P(p(n))=5 is 2^35 * 3^2 * 7^6 (17 digits).
%C A003001 The largest p(n) with P(p(n))=4 is 2^59 * 3^5 * 7^2 (22 digits).
%C A003001 The largest p(n) with P(p(n))=3 is 2^4 * 3^17 * 7^38 (42 digits).
%C A003001 The largest p(n) with P(p(n))=2 is 2^25 * 3^227 * 7^28 (140 digits).
%C A003001 All p(n) between 10^140 and 10^20000 have a persistence of 1, meaning they contain a 0 digit. (End)
%C A003001 _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
%C A003001 From _A.H.M. Smeets_, Nov 16 2018: (Start)
%C A003001 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.
%C A003001 For any multiplicative persistence, except multiplicative persistence 2, the set of class representative numbers with that multiplicative persistence is conjectured to be finite.
%C A003001 Each class representative number represents an infinite set of numbers with the same multiplicative persistence.
%C A003001 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:
%C A003001 final digit
%C A003001 mp total 0 1 2 3 4 5 6 7 8 9
%C A003001 ====================================================
%C A003001 0 10 1 1 1 1 1 1 1 1 1 1
%C A003001 1 10 1 1 1 1 1 1 1 1 1 1
%C A003001 2 inf inf 0 4 0 1 1 5 0 7 0
%C A003001 3 12199 12161 0 8 0 3 3 8 0 16 0
%C A003001 4 408 342 0 14 0 5 4 19 0 24 0
%C A003001 5 151 88 0 9 0 1 3 37 0 13 0
%C A003001 6 41 24 0 1 0 0 0 14 0 2 0
%C A003001 7 13 9 0 0 0 0 0 4 0 0 0
%C A003001 8 8 7 0 0 0 0 0 1 0 0 0
%C A003001 9 5 5 0 0 0 0 0 0 0 0 0
%C A003001 10 2 2 0 0 0 0 0 0 0 0 0
%C A003001 11 2 2 0 0 0 0 0 0 0 0 0
%C A003001 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).
%C A003001 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.
%C A003001 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).
%C A003001 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)
%C A003001 From _Tim Peters_, Sep 19 2023: (Start)
%C A003001 New lower bound: if a(12) exists, it must be > 2.67*10^30000. It continues to be the case that the digit products for all candidates with at least 20000 digits (roughly where the last long run reported here stopped) contain a zero digit, so the candidates all have persistence 2. More, the digit products all contain at least one zero in their last 306 digits. An extreme is the digit product 2^13802 * 3^16807 * 7^1757. That has 13659 decimal digits, 1335 of which are zeros. It ends with a zero followed by 305 nonzero digits. So to confirm that the large candidates with no more than 30000 digits have persistence 2, it would suffice to compute digit products modulo 10^306.
%C A003001 Note: by "candidate" I mean a digit string matching one of these eight (pairwise disjoint) simple regular expressions. Each such string gives the smallest integer with its digit product (and viewing the empty string as having digit product 1), and their union covers all digit products that don't end with a zero.
%C A003001 7* 8* 9*
%C A003001 2 7* 8* 9*
%C A003001 3 7* 8* 9*
%C A003001 4 7* 8* 9*
%C A003001 5 5* 7* 9*
%C A003001 6 7* 8* 9*
%C A003001 26 7* 8* 9*
%C A003001 35 5* 7* 9*
%C A003001 There are (8*N^2 + 13*N + 6)*(N + 1)/6 such strings with no more than N digits. A long computer run checked N=30000, a bit over 36*10^12 candidates. The smallest candidate with more than 30000 digits is > 2.67*10^30000, which is the smallest remaining possibility for a(12). (End)
%D A003001 Alex Bellos, Here's Looking at Euclid: A Surprising Excursion Through the Astonishing World of Math, Free Press, 2010, page 176.
%D A003001 M. Gardner, Fractal Music, Hypercards and More, Freeman, NY, 1991, pp. 170, 186.
%D A003001 R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section F25.
%D A003001 C. A. Pickover, Wonders of Numbers, "Persistence", Chapter 28, Oxford University Press NY 2001.
%D A003001 Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 66.
%D A003001 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A003001 James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 35.
%D A003001 David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 78.
%H A003001 Edson de Faria and Charles Tresser, <a href="http://arxiv.org/abs/1307.1188">On Sloane's persistence problem</a>, arXiv preprint arXiv:1307.1188 [math.DS], 2013.
%H A003001 Edson de Faria and Charles Tresser, <a href="http://dx.doi.org/10.1080/10586458.2014.910849">On Sloane's persistence problem</a>, Experimental Math., 23 (No. 4, 2014), 363-382.
%H A003001 Mark R. Diamond, <a href="http://web.archive.org/web/20160313225019/http://markdiamond.com.au/download/joous-3-1-1.pdf">Multiplicative persistence base 10: some new null results</a>, 2011.
%H A003001 Shyam Sunder Gupta, <a href="https://doi.org/10.1007/978-981-97-2465-9_1">Digital Root Wonders</a>, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 1, 1-28.
%H A003001 Brady Haran and Matt Parker, <a href="https://www.youtube.com/watch?v=Wim9WJeDTHQ">What's special about 277777788888899?</a>, Numberphile video, 2019.
%H A003001 T. Lamont-Smith, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Lamont/lamont5.html">Multiplicative Persistence and Absolute Multiplicative Persistence</a>, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
%H A003001 Kevin McElwee, <a href="https://medium.com/@kevinrmcelwee/multiplicative-persistence-is-solved-1937692b26cc">An algorithm for multiplicative persistence research</a>, Jul 13 2019.
%H A003001 S. Perez and R. Styer, <a href="http://www41.homepage.villanova.edu/robert.styer/MultiplicativePersistence/PersistenceStephPerezJournalArtAug2013.pdf">Persistence: A Digit Problem</a>.
%H A003001 W. Schneider, <a href="http://web.archive.org/web/2004/www.wschnei.de/digit-related-numbers/persistence.html">The Persistence of a Number</a>.
%H A003001 N. J. A. Sloane, <a href="http://neilsloane.com/doc/persistence.html">The persistence of a number</a>, J. Recreational Math., 6 (1973), 97-98.
%H A003001 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>.
%H A003001 Wikipedia, <a href="http://en.wikipedia.org/wiki/Persistence_of_a_number">Persistence of a number</a>.
%H A003001 Susan Worst, <a href="/A003001/a003001.pdf">Multiplicative persistence of base four numbers</a>. [Scanned copy of manuscript and correspondence, May 1980]
%e A003001 77 -> 49 -> 36 -> 18 -> 8 has persistence 4.
%t A003001 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 *)
%o A003001 (PARI) persistence(x)={my(y=digits(x),c=0);while(#y>1,y=digits(vecprod(y));c++);return(c)}
%o A003001 firstTermsA003001(U)={my(ans=vector(U),k=(U>1),z);while(k+1<=U,if(persistence(z)==k,ans[k++]=z);z++);return(ans)}
%o A003001 \\ Finds the first U terms (is slow); _R. J. Cano_, Sep 11 2016
%Y A003001 Cf. A031346 (persistence), A133500 (powertrain), A133048 (powerback), A006050, A007954, A031286, A031347, A033908, A046511, A121105-A121111.
%K A003001 nonn,nice,base,more,hard
%O A003001 0,2
%A A003001 _N. J. A. Sloane_