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

1,1

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
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

Adjacent sequences: A359095 A359096 A359097 * A359099 A359100 A359101

Keyword

nonn,base,easy

Author

Charles R Greathouse IV, Jan 02 2023

Status

approved