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%I #7 Jan 24 2023 10:28:39
%S 11,12,13,14,15,16,17,18,19,21,22,23,24,25,26,27,28,29,31,32,33,34,35,
%T 36,37,38,39,41,42,43,44,45,46,47,48,49,51,52,53,54,55,56,57,58,59,61,
%U 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
%N Numbers with exactly two nonzero decimal digits and not ending with 0.
%C 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 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.
%H 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 Numbers of the form a*10^b + c where 0 < a,c < 10 and b > 0.
%o (PARI) a(n)=my(t=divrem(n-1,81)); 10*(t[2]\9+1)*10^t[1]+t[2]%9+1
%Y Cf. A359651, A359098. Subsequence of A038687.
%K nonn,base,easy
%O 1,1
%A _Charles R Greathouse IV_, Jan 20 2023
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