A031346 - OEIS (original) (raw)

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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.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 35.

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](/A031346 "Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.") := 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](/A031346 "Multiplicative persistence: number of iterations of "multiply digits" needed to reach a number < 10.")(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

mp = 0

while n > 9:

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

mp += 1

return mp

(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).