OFFSET
1,1
COMMENTS
Bugeaud proves that the largest prime factor in a(n) increases without bound; in particular, for any e > 0 and all large n, the largest prime factor in a(n) is (1-e) * log log a(n) * log log log a(n) / log log log log a(n). So the largest prime factor in a(n) is more than k log n log log n/log log log n for any k < 1/3 and large enough n.
It appears that a(5177) = 8192 is the last 2-smooth member, a(26023) = 98304 is the last 3- and 5-smooth member, a(140723) = 16003008 is the last 7-smooth member, a(232305) = 100029006 is the last 11-smooth member, and a(419007) = 3009009003 is the last 13- and 17-smooth member.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Yann Bugeaud, On the digital representation of integers with bounded prime factors, Osaka J. Math. 55 (2018), 315-324; arXiv:1609.07926 [math.NT], 2016.
FORMULA
a(n) is roughly 10^(k*n^(1/3)), where k = (2/9)^(1/3)/3 = 0.2019....
MATHEMATICA
Select[Range[1111, 1199], And[Mod[#, 10] != 0, Total@ Most@ DigitCount[#] == 4] &] (* Michael De Vlieger, Jan 03 2023 *)
PROG
(PARI) list(lim)=my(v=List()); for(d=4, #Str(lim\=1), my(A=10^(d-1)); forstep(a=A, 9*A, A, for(i=2, d-2, my(B=10^i); forstep(b=a+B, a+9*B, B, for(j=1, i-1, my(C=10^j); forstep(c=b+C, b+9*C, C, for(d=c+1, c+9, if(d>lim, return(Vec(v))); listput(v, d)))))))); Vec(v)
(Python)
from itertools import count, islice
def A359098_gen(): # generator of terms
for a in count(3):
a10 = 10**a
for ad in range(1, 10):
for b in range(2, a):
b10 = 10**b
for bd in range(1, 10):
for c in range(1, b):
c10 = 10**c
for cd in range(1, 10):
for dd in range(1, 10):
yield ad*a10+bd*b10+cd*c10+dd
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Charles R Greathouse IV, Jan 02 2023
STATUS
approved