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Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.
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%I #58 Nov 15 2022 11:45:44

%S 0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,1,1,1,1,

%T 2,2,2,2,2,3,1,1,1,2,2,2,2,3,2,3,1,1,2,2,2,3,2,3,2,3,1,1,2,2,2,2,3,2,

%U 3,3,1,1,2,2,3,3,2,4,3,3,1,1,2,2,2,2,3,3,3,3,1,1,2,3,3,3,3,3,3,2

%N Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.

%D M. Gardner, Fractal Music, Hypercards and More Mathematical Recreations from Scientific American, Persistence of Numbers, pp. 120-1; 186-7, W. H. Freeman NY 1992.

%H T. D. Noe, <a href="/A031346/b031346.txt">Table of n, a(n) for n = 0..10000</a>

%H Gabriel Bonuccelli, Lucas Colucci, and Edson de Faria, <a href="https://arxiv.org/abs/2009.01114">On the Erdős-Sloane and Shifted Sloane Persistence</a>, arXiv:2009.01114 [math.NT], 2020.

%H Eric Brier, Christophe Clavier, Linda Gutsche and David Naccache, <a href="https://arxiv.org/abs/2110.04263">The Multiplicative Persistence Conjecture Is True for Odd Targets</a>, arXiv:2110.04263 [math.NT], 2021.

%H M. R. Diamond, <a href="http://www.markdiamond.com.au/download/joous-3-1-1.pdf">Multiplicative persistence base 10: some new null results</a>.

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>

%F Probably bounded, see A003001. - _Charles R Greathouse IV_, Nov 15 2022

%e For n = 999: A007954(999) = 729, A007954(729) = 126, A007954(126) = 12 and A007954(12) = 2. The fourth iteration of "multiply digits" yields a single-digit number, so a(999) = 4. - _Felix Fröhlich_, Jul 17 2016

%p A007954 := proc(n) return mul(d, d=convert(n, base, 10)): end: A031346 := proc(n) local k,m: k:=0:m:=n: while(length(m)>1)do m:=A007954(m):k:=k+1: od: return k: end: seq(A031346(n),n=0..100); # _Nathaniel Johnston_, May 04 2011

%t Table[Length[NestWhileList[Times@@IntegerDigits[#]&,n,#>=10&]],{n,0,100}]-1 (* _Harvey P. Dale_, Aug 27 2016 *)

%o (Python)

%o from operator import mul

%o from functools import reduce

%o def A031346(n):

%o mp = 0

%o while n > 9:

%o n = reduce(mul, (int(d) for d in str(n)))

%o mp += 1

%o return mp

%o # _Chai Wah Wu_, Aug 23 2014

%o (PARI) a007954(n) = my(d=digits(n)); prod(i=1, #d, d[i])

%o a(n) = my(k=n, i=0); while(#Str(k) > 1, k=a007954(k); i++); i \\ _Felix Fröhlich_, Jul 17 2016

%o (Magma) f:=func<n|&*Intseq(n)>; a:=[]; for n in [0..100] do s:=0; k:=n; while k ge 10 do s:=s+1; k:=f(k); end while; Append(~a,s); end for; a; // _Marius A. Burtea_, Jan 12 2020

%Y Cf. A007954 (product of decimal digits of n).

%Y Cf. A010888 (additive digital root of n).

%Y Cf. A031286 (additive persistence of n).

%Y Cf. A031347 (multiplicative digital root of n).

%Y Cf. A263131 (ordinal transform).

%Y Cf. A003001.

%K nonn,easy,base

%O 0,26

%A _Eric W. Weisstein_