OFFSET
2,1
COMMENTS
From Zachary DeStefano, May 17 2023: (Start)
There is a strong linear relationship between n^(n / phi(n)) and a(n) (see A000010 for phi(n)) which results from the final digit falling into subgroups of Z/nZ during split-and-multiply steps. This explains why a(n) is significantly smaller for prime n and significantly larger when n contains several small prime factors (ex. 2 * 3 * 5 = 30) (End)
LINKS
Zachary DeStefano and Tim Peters, Table of n, a(n) for n = 2..119
Michael S. Branicky, Python program
Michael De Vlieger, Plot of digit d in sequence S_n(m) in black at (x, y) = (m, d) for bases n = 2..12 as labeled, m = 1..1000, and digits d = 0..n-1, 12X exaggeration. This sequence shows the first occasion of a vertical black bar in base n.
Zachary DeStefano and Tim Peters, Known terms and bounds
EXAMPLE
To reach the digits 0 though 9 in base 10 from 17117:
171*17 -> 290*7 -> 203*0 -> 0
1711*7 -> 1197*7 -> 837*9 -> 7*533 -> 373*1 -> 37*3 -> 1*11 -> 1*1 -> 1
171*17 -> 2*907 -> 1*814 -> 8*14 -> 1*12 -> 1*2 -> 2
1*7117 -> 711*7 -> 49*77 -> 377*3 -> 113*1 -> 1*13 -> 1*3 -> 3
171*17 -> 2*907 -> 1*814 -> 8*14 -> 11*2 -> 2*2 -> 4
1711*7 -> 1197*7 -> 837*9 -> 75*33 -> 247*5 -> 1*235 -> 23*5 -> 1*15 -> 1*5 -> 5
17*117 -> 19*89 -> 169*1 -> 16*9 -> 1*44 -> 4*4 -> 1*6 -> 6
1711*7 -> 1197*7 -> 837*9 -> 7*533 -> 37*31 -> 11*47 -> 51*7 -> 3*57 -> 17*1 -> 1*7 -> 7
17*117 -> 1*989 -> 98*9 -> 88*2 -> 1*76 -> 7*6 -> 4*2 -> 8
1*7117 -> 711*7 -> 49*77 -> 377*3 -> 113*1 -> 11*3 -> 3*3 -> 9
MATHEMATICA
Table[Catch[Monitor[Do[(Set[c, Count[Union@Flatten[#], _?(# < b &)]]; If[c == b, Throw[i]]) &@ NestWhileList[Flatten@ Map[Function[w, Array[If[And[#[[-1, 1]] == 0, Length[#[[-1]]] > 1], Nothing, Times @@ Map[FromDigits[#, b] &, #]] &@ TakeDrop[w, #] &, Length[w] - 1]][IntegerDigits[#, b]] &, #] &, {i}, Length[#] > 0 &], {i, 0, Infinity}], {b, i, c}]], {b, 2, 6}] (* Michael De Vlieger, Apr 04 2023, with Monitor to show progress *)
PROG
(Python)
from itertools import count
from sympy.ntheory import digits
from functools import lru_cache
def fd(d, b): # from_digits
return sum(di*b**i for i, di in enumerate(d[::-1]))
@lru_cache(maxsize=None)
def f(n, b):
if n < b: return {n}
s = digits(n, b)[1:]
return {e for i in range(1, len(s)) if s[i]!=0 or i==len(s)-1 for e in f(fd(s[:i], b)*fd(s[i:], b), b)}
def a(n, printat=False):
return next(k for k in count(1) if len(f(k, n))==n)
print([a(n) for n in range(2, 18)]) # Michael S. Branicky, Apr 04 2023
(Python) # see link for a version that is faster and uses less memory
CROSSREFS
KEYWORD
nonn,base
AUTHOR
N. J. A. Sloane, Apr 04 2023, based on an email from Zachary DeStefano
EXTENSIONS
a(21)-a(29) from Michael S. Branicky, Apr 04 2023
a(30)-a(41) from Zachary DeStefano, Apr 05 2023
STATUS
approved