A359651 Numbers with exactly three nonzero decimal digits and not ending with 0.

111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 128, 129, 131, 132, 133, 134, 135, 136, 137, 138, 139, 141, 142, 143, 144, 145, 146, 147, 148, 149, 151, 152, 153, 154, 155, 156, 157, 158, 159, 161, 162, 163, 164, 165, 166, 167, 168, 169, 171, 172, 173, 174, 175

Offset: 1

Author

Charles R Greathouse IV, Jan 09 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(1293) = 4096 is the largest power of 2 in the sequence, a(1349) = 4608 is the largest 3-smooth number in this sequence, a(1598) = 6075 is the largest 5-smooth number in this sequence, a(5746) = 500094 is the largest 7- and 11-smooth number in this sequence, a(9158) = 5010005 is the largest 13-smooth member in this sequence, etc.

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.

Crossrefs

Cf. A359098.

Programs

Mathematica

Select[Range[111,175],Length[Select[IntegerDigits[#],Positive]]==3&&Mod[#,10]!=0 &] (* Stefano Spezia, Jan 15 2023 *)

PARI

list(lim)=my(v=List()); for(d=3, #Str(lim\=1), my(A=10^(d-1)); forstep(a=A, 9*A, A, for(i=1, d-2, my(B=10^i); forstep(b=a+B, a+9*B, B, for(n=b+1, b+9, if(n>lim, return(Vec(v))); listput(v, n)))))); Vec(v)