A245760 - OEIS

%I #16 Nov 13 2021 04:45:37

%S 0,1,1,1,1,1,1,2,1,2,2,1,2,2,2,2,2,2,2,2,2,2,3,2,2,3,3,3,3,2,3,2,3,3,

%T 3,2,3,2,3,3,3,3,3,3,3,3,3,2,3,3,3,4,3,3,3,3,3,3,3,3,4,3,3,3,3,3,3,4,

%U 4,3,3,3,3,3,4,3,4,3,3,3,4,3,4,3,3,4,3

%N Maximal multiplicative persistence of n in any base.

%C It has been conjectured that there is a maximum multiplicative persistence in a given base, but it is not known if this sequence is bounded.

%H Alois P. Heinz, <a href="/A245760/b245760.txt">Table of n, a(n) for n = 1..10000</a>

%e a(23)=3 since the persistence of 23 in base 6 is 3 (23 in base 6 is 35 / 3x5=15 / 15 in base 6 is 23 / 2x3=6 / 6 in base 6 is 10 / 1x0=0 which is a single digit). In any other base the persistence of 23 is 3 or less, therefore a(23)=3.

%e a(12)=1 since 12 does not have a multiplicative persistence greater than 1 in any base.

%p persistence:= proc(n,b) local i,m;

%p   m:= n;

%p   for i from 1 do

%p        m:= convert(convert(m,base,b),`*`);

%p      if m < b then return i fi

%p   od:

%p end proc:

%p A:= n -> max(seq(persistence(n,b),b=2..n-1)):

%p 0, 1, seq(A(n),n=3..100); # _Robert Israel_, Jul 31 2014

%t persistence[n_, b_] := Module[{i, m}, m = n; For[i = 1, True, i++, m = Times @@ IntegerDigits[m, b]; If[m < b, Return [i]]]];

%t A[n_] := Max[Table[persistence[n, b], {b, 2, n-1}]];

%t Join[{0, 1}, Table[A[n], {n, 3, 100}]] (* _Jean-François Alcover_, Apr 30 2019, after _Robert Israel_ *)

%Y Cf. A003001, A031346, A064867, A064868, A046510.

%K nonn

%O 1,8

%A _Sergio Pimentel_, Jul 31 2014