A245760 Maximal multiplicative persistence of n in any base.

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, 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, 4, 3, 3, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 4, 3

Offset: 1

Author

Sergio Pimentel, Jul 31 2014

Keywords

• nonn

Comments

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.

In fact, Theorem 1 in Lamont-Smith paper implies that this sequence is unbounded. - Brendan Gimby, Jul 12 2025

Examples

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

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Tim Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.

Crossrefs

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

Programs

Maple

persistence:= proc(n,b) local i,m;
  m:= n;
  for i from 1 do
       m:= convert(convert(m,base,b),`*`);
     if m < b then return i fi
  od:
end proc:
A:= n -> max(seq(persistence(n,b),b=2..n-1)):
0, 1, seq(A(n),n=3..100); # Robert Israel, Jul 31 2014

Mathematica

persistence[n_, b_] := Module[{i, m}, m = n; For[i = 1, True, i++, m = Times @@ IntegerDigits[m, b]; If[m < b, Return [i]]]];
A[n_] := Max[Table[persistence[n, b], {b, 2, n-1}]];
Join[{0, 1}, Table[A[n], {n, 3, 100}]] (* Jean-François Alcover, Apr 30 2019, after Robert Israel *)