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A031346
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Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.
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40
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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
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OFFSET
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0,26
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REFERENCES
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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.
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LINKS
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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.
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
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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Table[Length[NestWhileList[Times@@IntegerDigits[#]&, n, #>=10&]], {n, 0, 100}]-1 (* Harvey P. Dale, Aug 27 2016 *)
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PROG
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(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
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CROSSREFS
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Cf. A010888 (additive digital root of n).
Cf. A031286 (additive persistence of n).
Cf. A031347 (multiplicative digital root of n).
Cf. A263131 (ordinal transform).
Sequence in context: A102675 A177849 A143544 * A335808 A087472 A172069
Adjacent sequences: A031343 A031344 A031345 * A031347 A031348 A031349
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KEYWORD
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nonn,easy,base
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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