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A031346 Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10. 35
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 (list; graph; refs; listen; history; text; internal format)
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

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

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

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

CROSSREFS

Cf. A010888 (additive digital root of n).

Cf. A031286 (additive persistence of n).

Cf. A031347 (multiplicative digital root of n).

Sequence in context: A102675 A177849 A143544 * A087472 A172069 A054348

Adjacent sequences:  A031343 A031344 A031345 * A031347 A031348 A031349

KEYWORD

nonn,easy,base

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified November 22 10:40 EST 2014. Contains 249805 sequences.