A359098 Numbers with exactly four nonzero decimal digits and not ending with 0.

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

Author

Charles R Greathouse IV, Jan 02 2023

Keywords

• nonn
• base
• easy

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.

Crossrefs

Cf. A358737.

Programs

Mathematica

Select[Range[1111, 1199], And[Mod[#, 10] != 0, Total@ Most@ DigitCount[#] == 4] &] (* Michael De Vlieger, Jan 03 2023 *)

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

Formula

a(n) is roughly 10^(k*n^(1/3)), where k = (2/9)^(1/3)/3 = 0.2019....