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

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, 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, 12, 9, 9, 6, 3, 13, 5, 10, 11, 7, 10, 9, 3, 14, 14, 7

Offset

1,1

Comments

a(n) is the smallest base in which n has multiplicative persistence A245760(n).

Links

Brendan Gimby, Table of n, a(n) for n = 1..10000

Brendan Gimby, Tools for finding numbers with large persistence.

Tim Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.

N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.

Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Example

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.
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.

Mathematica

mp[n_, b_] := Module[{c = 0, cur = n}, While[cur >= b, cur = Times @@ IntegerDigits[cur, b]; c++  ]; c ];
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]]]]]  ];
Array[a, 82] (* James C. McMahon, Jul 17 2025 *)

Prog

(Python)
from math import prod
from sympy.ntheory.digits import digits
def mp(n, b): # multiplicative persistence of n in base b [from Michael S. Branicky in A330152]
    c = 0
    while n >= b:
        n, c = prod(digits(n, b)[1:]), c+1
    return c
def a(n):
    ps = list((mp(n, b) for b in range(2, max(3, n))))
    return ps.index(max(ps)) + 2
print([a(n) for n in range(1, 60)])

Crossrefs

Cf. A245760, A330152.

Sequence in context: A066088 A139326 A029243 * A306240 A109829 A054125

Adjacent sequences: A385927 A385928 A385929 * A385931 A385932 A385933

Keyword

nonn

Author

Brendan Gimby, Jul 12 2025

Status

approved