login
A359098
Numbers with exactly four nonzero decimal digits and not ending with 0.
4
1111, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 1138, 1139, 1141, 1142, 1143, 1144, 1145, 1146, 1147, 1148, 1149, 1151, 1152, 1153, 1154, 1155, 1156, 1157, 1158, 1159, 1161, 1162, 1163, 1164, 1165, 1166, 1167
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
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
A359098_list = list(islice(A359098_gen(), 30)) # Chai Wah Wu, Jan 03 2023
CROSSREFS
Cf. A358737.
Sequence in context: A290848 A290556 A343038 * A218042 A176670 A072434
KEYWORD
nonn,base,easy
AUTHOR
STATUS
approved