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A031347
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Multiplicative digital root of n (keep multiplying digits of n until reaching a single digit).
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67
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0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 2, 4, 6, 8, 0, 2, 4, 6, 8, 0, 3, 6, 9, 2, 5, 8, 2, 8, 4, 0, 4, 8, 2, 6, 0, 8, 6, 6, 8, 0, 5, 0, 5, 0, 0, 0, 5, 0, 0, 0, 6, 2, 8, 8, 0, 8, 8, 6, 0, 0, 7, 4, 2, 6, 5, 8, 8, 0, 8, 0, 8, 6, 8, 6, 0, 6
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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COMMENTS
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More precisely, a(n) = 0 asymptotically almost surely, namely, among others, for all numbers n which have a digit '0', and as n has more and more digits, it becomes increasingly less probable that no digit is equal to zero. (The set A011540 has density 1.) Thus the density of numbers for which a(n) > 0 is zero, although this happens for infinitely many numbers, for example all repunits n = (10^k - 1)/9 = A002275(k). - M. F. Hasler, Oct 11 2015
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LINKS
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FORMULA
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a(n) = d in {1, ..., 9} if (but not only if) n = (10^k - 1)/9 + (d - 1)*10^m = A002275(k) + (d - 1)*A011557(m) for some k > m >= 0. - M. F. Hasler, Oct 11 2015
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MAPLE
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MATHEMATICA
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mdr[n_] := NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]; Table[ mdr[n], {n, 0, 104}] (* Robert G. Wilson v, Aug 04 2006 *)
Table[NestWhile[Times@@IntegerDigits[#] &, n, # > 9 &], {n, 0, 90}] (* Harvey P. Dale, Mar 10 2019 *)
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PROG
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(PARI) A031347(n)=local(resul); if(n<10, return(n) ); resul = n % 10; n = (n - n%10)/10; while( n > 0, resul *= n %10; n = (n - n%10)/10; ); return(A031347(resul))
(PARI) A031347(n)={while(n>9, n=prod(i=1, #n=digits(n), n[i])); n} \\ M. F. Hasler, Dec 07 2014
(Haskell)
a031347 = until (< 10) a007954
(Python)
from operator import mul
from functools import reduce
while n > 9:
n = reduce(mul, (int(d) for d in str(n)))
return n
(Python)
from math import prod
while n > 9: n = prod(map(int, str(n)))
return n
(Scala) def iterDigitProd(n: Int): Int = n.toString.length match {
case 1 => n
case _ => iterDigitProd(n.toString.toCharArray.map(_ - 48).scanRight(1)(_ * _).head)
}
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CROSSREFS
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KEYWORD
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nonn,base,easy,nice
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AUTHOR
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STATUS
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approved
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