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



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.


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

M. R. Diamond, Multiplicative persistence base 10: some new null results, http://www.markdiamond.com.au/download/joous-3-1-1.pdf.

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).


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

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


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


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 *)


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




N. J. A. Sloane.



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Last modified December 19 20:36 EST 2014. Contains 252239 sequences.