%I #68 Apr 17 2024 08:00:47

%S 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,

%T 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,

%U 6,0,0,7,4,2,6,5,8,8,0,8,0,8,6,8,6,0,6

%N Multiplicative digital root of n (keep multiplying digits of n until reaching a single digit).

%C a(n) = 0 for almost all n. - _Charles R Greathouse IV_, Oct 02 2013

%C 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

%H T. D. Noe, <a href="/A031347/b031347.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultiplicativeDigitalRoot.html">Multiplicative Digital Root</a>.

%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>

%F 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

%p A007954 := proc(n) return mul(d, d=convert(n,base,10)): end: A031347 := proc(n) local m: m:=n: while(length(m)>1)do m:=A007954(m): od: return m: end: seq(A031347(n),n=0..100); # _Nathaniel Johnston_, May 04 2011

%t mdr[n_] := NestWhile[Times @@ IntegerDigits@# &, n, UnsameQ, All]; Table[ mdr[n], {n, 0, 104}] (* _Robert G. Wilson v_, Aug 04 2006 *)

%t Table[NestWhile[Times@@IntegerDigits[#] &, n, # > 9 &], {n, 0, 90}] (* _Harvey P. Dale_, Mar 10 2019 *)

%o (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))

%o for(n=1,80, print1(A031347(n),",")) \\ _R. J. Mathar_, May 23 2006

%o (PARI) A031347(n)={while(n>9,n=prod(i=1,#n=digits(n),n[i]));n} \\ _M. F. Hasler_, Dec 07 2014

%o (Haskell)

%o a031347 = until (< 10) a007954

%o -- _Reinhard Zumkeller_, Oct 17 2011, Sep 22 2011

%o (Python)

%o from operator import mul

%o from functools import reduce

%o def A031347(n):

%o while n > 9:

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

%o return n

%o # _Chai Wah Wu_, Aug 23 2014

%o (Python)

%o from math import prod

%o def A031347(n):

%o while n > 9: n = prod(map(int, str(n)))

%o return n

%o print([A031347(n) for n in range(100)]) # _Michael S. Branicky_, Apr 17 2024

%o (Scala) def iterDigitProd(n: Int): Int = n.toString.length match {

%o case 1 => n

%o case _ => iterDigitProd(n.toString.toCharArray.map(_ - 48).scanRight(1)(_ * _).head)

%o }

%o (0 to 99).map(iterDigitProd) // _Alonso del Arte_, Apr 11 2020

%Y Cf. A007954, A007953, A003001, A010888 (additive digital root of n), A031286 (additive persistence of n), A031346 (multiplicative persistence of n).

%Y Numbers having multiplicative digital roots 0-9: A034048, A002275, A034049, A034050, A034051, A034052, A034053, A034054, A034055, A034056.

%K nonn,base,easy,nice

%O 0,3

%A _Eric W. Weisstein_