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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. 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.

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


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 February 10 03:44 EST 2016. Contains 268147 sequences.