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

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

Author

Brendan Gimby, Jul 08 2025

Keywords

• nonn
• more

Comments

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

Examples

			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).
		

Links

• 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.

Crossrefs

Cf. A003001, A245760, A330152.

Programs

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):
    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)])

Extensions

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