cp's OEIS Frontend

A330152 Absolute multiplicative persistence: a(n) is the least number with multiplicative persistence n for some base b > 1.

%I A330152 #53 Jul 08 2025 22:24:47
%S A330152 0,2,8,23,52,127,218,412,542,692,1471,2064,2327,4739,13025,16213,
%T A330152 20388,45407,82605,123706,207778,323382,605338,905670,1033731,2041995,
%U A330152 3325970,4282238,7638962,9840138,10364329,26110715,40706834,57222153,82242809,97900397
%N A330152 Absolute multiplicative persistence: a(n) is the least number with multiplicative persistence n for some base b > 1.
%H A330152 Edson de Faria and Charles Tresser, <a href="http://arxiv.org/abs/1307.1188">On Sloane's persistence problem</a>, arXiv preprint arXiv:1307.1188 [math.DS], 2013.
%H A330152 Edson de Faria and Charles Tresser, <a href="http://dx.doi.org/10.1080/10586458.2014.910849">On Sloane's persistence problem</a>, Experimental Math., 23 (No. 4, 2014), 363-382.
%H A330152 Brendan Gimby, <a href="https://github.com/bgimby/cudf-persistence-tools/tree/main">Tools for finding numbers with large persistence</a>.
%H A330152 Brady Haran and Matt Parker, <a href="https://www.youtube.com/watch?v=Wim9WJeDTHQ">What's special about 277777788888899?</a>, Numberphile video, 2019.
%H A330152 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.
%H A330152 Stephanie Perez and Robert Styer, <a href="http://dx.doi.org/10.2140/involve.2015.8.439">Persistence: A Digit Problem</a>, Involve, Vol. 8 (2015), No. 3, 439-446.
%H A330152 N. J. A. Sloane, <a href="http://neilsloane.com/doc/persistence.html">The persistence of a number</a>, J. Recreational Math., 6 (1973), 97-98.
%H A330152 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultiplicativePersistence.html">Multiplicative Persistence</a>.
%e A330152 2 when represented in base 2 goes 10 -> 0 and has an absolute persistence of 1, so a(1) = 2.
%e A330152 8 when represented in base 3 goes 22 -> 11 -> 1 and has an absolute persistence of 2, so a(2) = 8.
%e A330152 23 when represented in base 6 goes 35 -> 23 -> 10 -> 1 and has absolute persistence of 3, so a(3) = 23 (Cf. A064867).
%e A330152 52 when represented in base 9 goes 57 -> 38 -> 26 -> 13 -> 3 and has absolute persistence of 4, so a(4) = 52 (Cf. A064868).
%o A330152 (Python)
%o A330152 from math import prod
%o A330152 from sympy.ntheory.digits import digits
%o A330152 def mp(n, b): # multiplicative persistence of n in base b
%o A330152     c = 0
%o A330152     while n >= b:
%o A330152         n, c = prod(digits(n, b)[1:]), c+1
%o A330152     return c
%o A330152 def a(n):
%o A330152     k = 0
%o A330152     while True:
%o A330152         if any(mp(k, b)==n for b in range(2, max(3, k))): return k
%o A330152         k += 1
%o A330152 print([a(n) for n in range(11)]) # _Michael S. Branicky_, Sep 17 2021
%Y A330152 Cf. A003001, A064867, A064868, A064869, A064870, A064871, A064872.
%K A330152 nonn,base
%O A330152 0,2
%A A330152 _Tim Lamont-Smith_, Nov 29 2019
%E A330152 a(19)-a(27) from _Giovanni Resta_, Jan 20 2020
%E A330152 a(28)-a(30) from _Michael S. Branicky_, Sep 17 2021
%E A330152 a(31)-a(35) from _Brendan Gimby_, Jul 08 2025