mutually-praising pairs

[Joshua Zucker]

On Tue, Mar 18, 2008 at 10:07 AM, Tanya Khovanova
<mathoflove-seqfan at yahoo.com> wrote:
>  Mutually-praising pairs.
>  130, 230, 430, 530, 630, 730, 1101, 2210, 10110, 11200, 23100
>
>  Comment: 1101 and 130 mutually describe each other: If you read 130 as
>  1 zero, 3 ones, 0 twos; you will get the description of 1101. If you
>  read 1101 as 1 zero, 1 one, 0 twos and 1 three; you will get the
>  description of 130.

Maybe I don't understand the definition - why are for example 12 and
110 not mutually praising?  And why not 32 and 11100?  Ah, I see,
perhaps leading zeros are not allowed.

Anyway, with my definition (allowing leading 0s), I get the following
as all the mutually praising pairs that have at least one element less
than or equal to 100000:
(1 1) (12 110) (32 11000) (42 101000) (52 1001000) (62 10001000) (110
12) (130 1101) (211 2100) (230 10110) (311 20100) (411 200100) (430
1001100) (511 2000100) (530 10010100) (611 20000100) (630 100100100)
(711 200000100) (730 1001000100) (1101 130) (1210 1210) (2020 2020)
(2100 211) (2210 11200) (10110 230) (11000 32) (11200 2210) (20100
311) (21200 21200) (23100 211100) (43100 21011000) (53100 210101000)
(63100 2101001000)

and thus the following as the start of your sequence:
1 12 32 42 52 62 110 130 211 230 311 411 430 511 530 611 630 711 730
1101 1210 2020 2100 2210 10110 11000 11200 20100 21200 23100 43100
53100 63100

I also include in the above the autobiographical numbers which it
appears you want to have excluded.

If I go to what appears to be your definition, excluding leading 0s
from the numbers, then I get
130 230 430 530 630 730 1101 1210 2020 2210 10110 11200 21200 23100
43100 53100 63100 211100 230100 311100 411100 430100 511100 530100
611100 630100 711100 1001100 2300100 3211000 4211000 4300100 5211000
5300100 6211000 6300100
but that includes the autobiographical numbers as well.

--Joshua Zucker