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

1,8

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.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Example

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.

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 *)

Crossrefs

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

Sequence in context: A264998 A118916 A107800 * A085761 A352884 A102382

Adjacent sequences: A245757 A245758 A245759 * A245761 A245762 A245763

Keyword

nonn

Author

Sergio Pimentel, Jul 31 2014

Status

approved