A181862 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.

1, 3, 9, 10, 11, 12, 18, 21, 27, 30, 33, 36, 41, 45, 54, 63, 72, 81, 90, 99, 100, 101, 102, 108, 110, 111, 117, 120, 123, 126, 132, 135, 144, 153, 162, 171, 180, 198, 201, 207, 210, 216, 225, 231, 234, 243, 252, 261, 270, 297, 300, 303, 306, 315, 324, 330, 333, 342, 351, 360, 396, 405, 410

Offset

1,2

Comments

Positive integers m such that A007953(m) = A077196(m).

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.

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

Links

Jason Yuen, Table of n, a(n) for n = 1..10000

Trevor Clokie, Thomas F. Lidbetter, Antonio Molina Lovett, Jeffrey Shallit, and Leon Witzman, Computational Aspects of Sturdy and Flimsy Numbers, arXiv:2002.02731 [cs.DS], 2020.

Example

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.
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.
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.

Crossrefs

Cf. A125121, A181863.

Sequence in context: A273893 A228165 A114022 * A329560 A330296 A043048

Adjacent sequences: A181859 A181860 A181861 * A181863 A181864 A181865

Keyword

nonn,base

Author

Max Alekseyev, Nov 14 2010

Status

approved