login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A330152
Absolute multiplicative persistence: a(n) is the least number with multiplicative persistence n for some base b > 1.
0
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
OFFSET
0,2
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.
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
EXAMPLE
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).
PROG
(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
KEYWORD
nonn,base,more
AUTHOR
Tim Lamont-Smith, Nov 29 2019
EXTENSIONS
a(19)-a(27) from Giovanni Resta, Jan 20 2020
a(28)-a(30) from Michael S. Branicky, Sep 17 2021
STATUS
approved