A330152 -id:A330152 - cp's OEIS frontend

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

Brendan Gimby, Jul 12 2025

nonn

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

			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.

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.

Cf. A245760, A330152.

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 *)

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)])

A385727 Minimum base in which the least number with absolute multiplicative persistence n achieves such persistence.

Original entry on oeis.org

0, 2, 3, 6, 9, 13, 17, 23, 26, 29, 41, 53, 53, 73, 123, 159, 157, 251, 332, 491, 587, 691, 943, 1187, 1187, 1804, 2923, 2348, 6241, 3541, 3541, 7082, 7082, 14164, 10623, 14164, 28328, 56656, 98533

Offset: 0

Brendan Gimby, Jul 08 2025

a(n) is the minimum base in which the n-th term in A330152 achieves its absolute persistence.

			23 when represented in base 6 goes 35 -> 23 -> 10 -> 1, has absolute persistence of 3, and has smaller persistence in all smaller bases, so a(3) = 6 (Cf. A064867).
52 when represented in base 9 goes 57 -> 38 -> 26 -> 13 -> 3, has absolute persistence of 4, and has smaller persistence in all smaller bases, so a(4) = 9 (Cf. A064868).

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.

Cf. A003001, A245760, A330152.

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):
    k = 0
    while True:
        if b := next((b for b in range(2, max(3, k)) if mp(k, b)==n), 0): return b
        k += 1
print([a(n) for n in range(11)])

a(36)-a(38) from Jinyuan Wang, Jul 13 2025

cp's OEIS Frontend

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

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