A359983 Numbers with exactly two nonzero decimal digits and not ending with 0.

11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 201

Offset: 1

Author

Charles R Greathouse IV, Jan 20 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(49) = 64 is the largest power of 2 in the sequence, a(78) = 96 is the largest 3-smooth number in this sequence, a(113) = 405 is the largest 5-smooth number in this sequence, a(170) = 1008 is the largest 7- and 11-smooth number in this sequence, a(243) = 9009 is the largest 13-smooth number in this sequence, a(259) = 20007 is the largest 19-smooth number 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. A359651, A359098. Subsequence of A038687.

Programs

PARI

a(n)=my(t=divrem(n-1,81)); 10*(t[2]\9+1)*10^t[1]+t[2]%9+1

Formula

Numbers of the form a*10^b + c where 0 < a,c < 10 and b > 0.