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%I #23 Mar 26 2024 04:33:40
%S 1,3,9,10,11,12,18,21,27,30,33,36,41,45,54,63,72,81,90,99,100,101,102,
%T 108,110,111,117,120,123,126,132,135,144,153,162,171,180,198,201,207,
%U 210,216,225,231,234,243,252,261,270,297,300,303,306,315,324,330,333,342,351,360,396,405,410
%N Decimal sturdy numbers: positive integers m such that sum of digits of k * m for any positive integer k is at least the sum of digits of m.
%C Positive integers m such that A007953(m) = A077196(m).
%C All powers of 10 and many multiples of 3 are in this sequence, many prime numbers are not. Notable exceptions are the primes 11 and 41 that are in this sequence, and multiples of 3 like 6 and 15 that are not.
%C This suggests that a digit sum of 6 disqualifies a multiple of 3 from this sequence, not parity. A digit sum of 9, by contrast, ensures the number is in this sequence. - _Alonso del Arte_, Oct 02 2016
%H Jason Yuen, <a href="/A181862/b181862.txt">Table of n, a(n) for n = 1..10000</a>
%H Trevor Clokie, Thomas F. Lidbetter, Antonio Molina Lovett, Jeffrey Shallit, and Leon Witzman, <a href="https://arxiv.org/abs/2002.02731">Computational Aspects of Sturdy and Flimsy Numbers</a>, arXiv:2002.02731 [cs.DS], 2020.
%e 11 has a digit sum of 2. If a multiple of 11 exists with a digit sum of 1, that would mean a power of 10 is also a multiple of 11, which is absurd. Therefore 11 is in the sequence.
%e 12 = 2^2 * 3 has a digit sum of 3. In base 10, all multiples of 3 have a digital root of 3, 6 or 9, which means that a total digit sum of 1 or 2 is impossible for a multiple of 3. Therefore 12 is in the sequence.
%e 13 has a digit sum of 4. However, note that 7 * 11 * 13 = 1001, which has a digit sum of 2. So 13 is not in the sequence.
%Y Cf. A125121, A181863.
%K nonn,base
%O 1,2
%A _Max Alekseyev_, Nov 14 2010
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