cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

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%I A359098 #39 Jan 04 2023 14:32:38
%S A359098 1111,1112,1113,1114,1115,1116,1117,1118,1119,1121,1122,1123,1124,
%T A359098 1125,1126,1127,1128,1129,1131,1132,1133,1134,1135,1136,1137,1138,
%U A359098 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
%N A359098 Numbers with exactly four nonzero decimal digits and not ending with 0.
%C A359098 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.
%C A359098 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.
%H A359098 Michael De Vlieger, <a href="/A359098/b359098.txt">Table of n, a(n) for n = 1..10000</a>
%H A359098 Yann Bugeaud, <a href="https://arxiv.org/abs/1609.07926">On the digital representation of integers with bounded prime factors</a>, Osaka J. Math. 55 (2018), 315-324; arXiv:1609.07926 [math.NT], 2016.
%F A359098 a(n) is roughly 10^(k*n^(1/3)), where k = (2/9)^(1/3)/3 = 0.2019....
%t A359098 Select[Range[1111, 1199], And[Mod[#, 10] != 0, Total@ Most@ DigitCount[#] == 4] &] (* _Michael De Vlieger_, Jan 03 2023 *)
%o A359098 (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)
%o A359098 (Python)
%o A359098 from itertools import count, islice
%o A359098 def A359098_gen(): # generator of terms
%o A359098     for a in count(3):
%o A359098         a10 = 10**a
%o A359098         for ad in range(1,10):
%o A359098             for b in range(2,a):
%o A359098                 b10 = 10**b
%o A359098                 for bd in range(1,10):
%o A359098                     for c in range(1,b):
%o A359098                         c10 = 10**c
%o A359098                         for cd in range(1,10):
%o A359098                             for dd in range(1,10):
%o A359098                                 yield ad*a10+bd*b10+cd*c10+dd
%o A359098 A359098_list = list(islice(A359098_gen(),30)) # _Chai Wah Wu_, Jan 03 2023
%Y A359098 Cf. A358737.
%K A359098 nonn,base,easy
%O A359098 1,1
%A A359098 _Charles R Greathouse IV_, Jan 02 2023