A373117 Stable numbers on vertical blade (see the Example section for an explanation).
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 22, 30, 33, 40, 44, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 101, 102, 110, 111, 113, 121, 124, 131, 135, 141, 146, 151, 157, 161, 168, 171, 179, 181, 191, 200, 201, 202, 204, 212, 215, 220, 222, 226, 232, 237, 242, 248, 252, 259, 262, 272, 282, 292, 300, 303, 306, 311
Offset: 1
Examples
We place the k digits of a number in succession in k adjacent square boxes forming a rectangle of base k and height 1. We can only place a vertical blade under this rectangle in two ways: exactly between two boxes [positions (a) and (d) below] or exactly under the middle of a box [position (b) and (c) below]. If the blade is placed in position (a) between the two 1s of 11, the number 11 is stable. If the blade is placed in (b) exactly under the 0 of 101, the number 101 is stable. If the blade is placed in (c) exactly under the 1 of 10, the number is stable. If the blade is placed in (d) between the 0 and the 2 of 102, the number 102 is stable. For the last two examples, the balance of the number considered can be explained as follows: the distance which separates a digit from the blade comes into play - the more this distance increases, the heavier the digit in question is. So, the digit 1 of 102 weighs 2 in reality (weight*distance = 1*2 = 2). This quantity balances the influence of the 2 of 102 (for which weight*distance = 2*1 = 2 too). . .+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ .| | | | | | | | | | | | | | | | .+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ .| | 1 | 1 | | 1 | 0 | 1 | | 1 | 0 | | 1 | 0 | 2 | | .+---+---a---+---+---+-b-+---+---+-c-+---+---+---+---d---+---+ .| | | | | | | | | | | | | | | | .+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+ .
Links
- Eric Angelini, Balanced numbers, personal blog, May 2024.
- Hans Havermann, 10000-term b-file.