A031346 Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.

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, 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, 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

Offset

0,26

References

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.

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

Gabriel Bonuccelli, Lucas Colucci, and Edson de Faria, On the Erdős-Sloane and Shifted Sloane Persistence, arXiv:2009.01114 [math.NT], 2020.

Eric Brier, Christophe Clavier, Linda Gutsche and David Naccache, The Multiplicative Persistence Conjecture Is True for Odd Targets, arXiv:2110.04263 [math.NT], 2021.

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

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Eric Weisstein's World of Mathematics, Multiplicative Persistence

Formula

Probably bounded, see A003001. - Charles R Greathouse IV, Nov 15 2022

Example

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

Maple

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

Mathematica

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

Prog

(Python)
from operator import mul
from functools import reduce
def A031346(n):
    mp = 0
    while n > 9:
        n = reduce(mul, (int(d) for d in str(n)))
        mp += 1
    return mp
# Chai Wah Wu, Aug 23 2014
(PARI) a007954(n) = my(d=digits(n)); prod(i=1, #d, d[i])
a(n) = my(k=n, i=0); while(#Str(k) > 1, k=a007954(k); i++); i \\ Felix Fröhlich, Jul 17 2016
(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

Crossrefs

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

Cf. A010888 (additive digital root of n).

Cf. A031286 (additive persistence of n).

Cf. A031347 (multiplicative digital root of n).

Cf. A263131 (ordinal transform).

Cf. A003001.

Sequence in context: A102675 A177849 A143544 * A335808 A087472 A172069

Adjacent sequences: A031343 A031344 A031345 * A031347 A031348 A031349

Keyword

nonn,easy,base

Author

Eric W. Weisstein

Status

approved