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A385930 Minimum base in which n achieves its maximum multiplicative persistence.

%I A385930 #24 Jul 21 2025 13:15:07
%S A385930 2,2,2,2,2,2,2,3,2,4,4,2,5,4,4,6,3,5,5,6,6,5,6,5,3,3,7,6,6,4,8,6,6,6,
%T A385930 9,8,8,4,7,7,7,9,11,8,7,7,7,5,10,9,9,9,3,8,8,12,10,9,6,8,9,8,4,6,6,10,
%U A385930 12,9,9,6,3,13,5,10,11,7,10,9,3,14,14,7
%N A385930 Minimum base in which n achieves its maximum multiplicative persistence.
%C A385930 a(n) is the smallest base in which n has multiplicative persistence A245760(n).
%H A385930 Brendan Gimby, <a href="/A385930/b385930.txt">Table of n, a(n) for n = 1..10000</a>
%H A385930 Brendan Gimby, <a href="https://github.com/bgimby/cudf-persistence-tools/tree/main">Tools for finding numbers with large persistence</a>.
%H A385930 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.
%H A385930 N. J. A. Sloane, <a href="http://neilsloane.com/doc/persistence.html">The persistence of a number</a>, J. Recreational Math., 6 (1973), 97-98.
%H A385930 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>.
%e A385930 8 written in base 3 goes 22 -> 11 -> 1, so 8 has persistence 2 in base 3. Since 8 has lower persistence in all smaller bases and no larger persistence in any higher bases, a(8)=3.
%e A385930 In all bases, 12 has persistence 1 or zero. In base 2, 12 goes 1100 -> 0 where it has persistence 1. Thus a(12)=2.
%t A385930 mp[n_, b_] := Module[{c = 0, cur = n},While[cur >= b, cur = Times @@ IntegerDigits[cur, b]; c++  ]; c ];
%t A385930 a[n_] := Module[{bases, persist},bases = Range[2, Max[3, n] - 1];persist = mp[n, #] & /@ bases;If[persist == {}, 2, bases[[Position[persist, Max[persist]][[1, 1]]]]]  ];
%t A385930 Array[a,82] (* _James C. McMahon_, Jul 17 2025 *)
%o A385930 (Python)
%o A385930 from math import prod
%o A385930 from sympy.ntheory.digits import digits
%o A385930 def mp(n, b): # multiplicative persistence of n in base b [from _Michael S. Branicky_ in A330152]
%o A385930     c = 0
%o A385930     while n >= b:
%o A385930         n, c = prod(digits(n, b)[1:]), c+1
%o A385930     return c
%o A385930 def a(n):
%o A385930     ps = list((mp(n, b) for b in range(2, max(3, n))))
%o A385930     return ps.index(max(ps)) + 2
%o A385930 print([a(n) for n in range(1, 60)])
%Y A385930 Cf. A245760, A330152.
%K A385930 nonn
%O A385930 1,1
%A A385930 _Brendan Gimby_, Jul 12 2025