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

0, 2, 8, 23, 52, 127, 218, 412, 542, 692, 1471, 2064, 2327, 4739, 13025, 16213, 20388, 45407, 82605, 123706, 207778, 323382, 605338, 905670, 1033731, 2041995, 3325970, 4282238, 7638962, 9840138, 10364329, 26110715, 40706834, 57222153, 82242809, 97900397

Offset: 0

Author

Tim Lamont-Smith, Nov 29 2019

Keywords

• nonn
• base

Examples

			2 when represented in base 2 goes 10 -> 0 and has an absolute persistence of 1, so a(1) = 2.
8 when represented in base 3 goes 22 -> 11 -> 1 and has an absolute persistence of 2, so a(2) = 8.
23 when represented in base 6 goes 35 -> 23 -> 10 -> 1 and has absolute persistence of 3, so a(3) = 23 (Cf. A064867).
52 when represented in base 9 goes 57 -> 38 -> 26 -> 13 -> 3 and has absolute persistence of 4, so a(4) = 52 (Cf. A064868).
		

Links

• Edson de Faria and Charles Tresser, On Sloane's persistence problem, arXiv preprint arXiv:1307.1188 [math.DS], 2013.
• Edson de Faria and Charles Tresser, On Sloane's persistence problem, Experimental Math., 23 (No. 4, 2014), 363-382.
• Brendan Gimby, Tools for finding numbers with large persistence.
• Brady Haran and Matt Parker, What's special about 277777788888899?, Numberphile video, 2019.
• Tim Lamont-Smith, Multiplicative Persistence and Absolute Multiplicative Persistence, J. Int. Seq., Vol. 24 (2021), Article 21.6.7.
• Stephanie Perez and Robert Styer, Persistence: A Digit Problem, Involve, Vol. 8 (2015), No. 3, 439-446.
• N. J. A. Sloane, The persistence of a number, J. Recreational Math., 6 (1973), 97-98.
• Eric Weisstein's World of Mathematics, Multiplicative Persistence.

Crossrefs

Cf. A003001, A064867, A064868, A064869, A064870, A064871, A064872.

Programs

Python

from math import prod
from sympy.ntheory.digits import digits
def mp(n, b): # multiplicative persistence of n in base b
    c = 0
    while n >= b:
        n, c = prod(digits(n, b)[1:]), c+1
    return c
def a(n):
    k = 0
    while True:
        if any(mp(k, b)==n for b in range(2, max(3, k))): return k
        k += 1
print([a(n) for n in range(11)]) # Michael S. Branicky, Sep 17 2021

Extensions

a(19)-a(27) from Giovanni Resta, Jan 20 2020

a(28)-a(30) from Michael S. Branicky, Sep 17 2021

a(31)-a(35) from Brendan Gimby, Jul 08 2025