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A245760 Maximal multiplicative persistence of n in any base.

%I A245760 #24 Jul 17 2025 03:08:38
%S A245760 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 A245760 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 A245760 4,3,3,3,3,3,4,3,4,3,3,3,4,3,4,3,3,4,3
%N A245760 Maximal multiplicative persistence of n in any base.
%C A245760 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.
%C A245760 In fact, Theorem 1 in Lamont-Smith paper implies that this sequence is unbounded. - _Brendan Gimby_, Jul 12 2025
%H A245760 Alois P. Heinz, <a href="/A245760/b245760.txt">Table of n, a(n) for n = 1..10000</a>
%H A245760 Tim Lamont-Smith, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Lamont/lamont5.html">Multiplicative Persistence and Absolute Multiplicative Persistence</a>, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
%e A245760 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 A245760 a(12)=1 since 12 does not have a multiplicative persistence greater than 1 in any base.
%p A245760 persistence:= proc(n,b) local i,m;
%p A245760   m:= n;
%p A245760   for i from 1 do
%p A245760        m:= convert(convert(m,base,b),`*`);
%p A245760      if m < b then return i fi
%p A245760   od:
%p A245760 end proc:
%p A245760 A:= n -> max(seq(persistence(n,b),b=2..n-1)):
%p A245760 0, 1, seq(A(n),n=3..100); # _Robert Israel_, Jul 31 2014
%t A245760 persistence[n_, b_] := Module[{i, m}, m = n; For[i = 1, True, i++, m = Times @@ IntegerDigits[m, b]; If[m < b, Return [i]]]];
%t A245760 A[n_] := Max[Table[persistence[n, b], {b, 2, n-1}]];
%t A245760 Join[{0, 1}, Table[A[n], {n, 3, 100}]] (* _Jean-François Alcover_, Apr 30 2019, after _Robert Israel_ *)
%Y A245760 Cf. A003001, A031346, A064867, A064868, A046510.
%K A245760 nonn
%O A245760 1,8
%A A245760 _Sergio Pimentel_, Jul 31 2014