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A003001 Smallest number of multiplicative persistence n.
(Formerly M4687)
51
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)

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.

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

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,hard

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified August 18 10:51 EDT 2017. Contains 290710 sequences.