cp's OEIS Frontend

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

%I A385727 #22 Jul 21 2025 13:14:55
%S A385727 0,2,3,6,9,13,17,23,26,29,41,53,53,73,123,159,157,251,332,491,587,691,
%T A385727 943,1187,1187,1804,2923,2348,6241,3541,3541,7082,7082,14164,10623,
%U A385727 14164,28328,56656,98533
%N A385727 Minimum base in which the least number with absolute multiplicative persistence n achieves such persistence.
%C A385727 a(n) is the minimum base in which the n-th term in A330152 achieves its absolute persistence.
%H A385727 Brendan Gimby, <a href="https://github.com/bgimby/cudf-persistence-tools/tree/main">Tools for finding numbers with large persistence</a>.
%H A385727 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.
%e A385727 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).
%e A385727 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).
%o A385727 (Python)
%o A385727 from math import prod
%o A385727 from sympy.ntheory.digits import digits
%o A385727 def mp(n, b): # multiplicative persistence of n in base b [from _Michael S. Branicky_ in A330152]
%o A385727     c = 0
%o A385727     while n >= b:
%o A385727         n, c = prod(digits(n, b)[1:]), c+1
%o A385727     return c
%o A385727 def a(n):
%o A385727     k = 0
%o A385727     while True:
%o A385727         if b := next((b for b in range(2, max(3, k)) if mp(k, b)==n), 0): return b
%o A385727         k += 1
%o A385727 print([a(n) for n in range(11)])
%Y A385727 Cf. A003001, A245760, A330152.
%K A385727 nonn,more
%O A385727 0,2
%A A385727 _Brendan Gimby_, Jul 08 2025
%E A385727 a(36)-a(38) from _Jinyuan Wang_, Jul 13 2025