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

0,2

Comments

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

Links

Table of n, a(n) for n=0..38.

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.

Example

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).

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):
    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)])

Crossrefs

Cf. A003001, A245760, A330152.

Sequence in context: A383229 A190772 A130673 * A374702 A306777 A062891

Adjacent sequences: A385724 A385725 A385726 * A385728 A385729 A385730

Keyword

nonn,more

Author

Brendan Gimby, Jul 08 2025

Extensions

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

Status

approved