Question related to sequence A071267

[Joshua Zucker]

Hi Diana and all,
Murthy's sequences often have errors.  Here, among other problems, the
definition is "numbers which ..." but then shouldn't they be SORTED?

I don't quite agree with your list, either, though.  Here's what I
think, based purely on brute-force with a computer, using as "seeds"
all the numbers up to 100000:

1 2 3 4 5 6 7 8 9 11 22 33 44 55 66 77 88 99 110 111 121 132 143 154
165 176 187 222 333 444 555 666 777 888 999 1110 1111 1221 1332 1443
1554 1665 1776 1887 1998 2109 2220 2222 2331 2442 2553 2664 2775 2886
3108 3330 3333 3552 3774 3996 4218 4440 4444 4662 4884 5106 5328 5555
6666 7777 8888 9999 11110 11111 12221 13332 14443 15554 16665 17776
18887 19998 21109 22220 22222 23331 24442 25553 26664 27775 28886
29997

which leaves me wondering, for instance, why I got 29997 but not 2997
in there.  But now I see it's a nice little bit of combinatorics: 108
goes to 1998 (six permutations, so effectively each spot is
(1+0+8+1+0+8) so we get 1800 + 180 + 18), while 1008 goes to 29997
(twelve permutations, so each spot is (1+0+0+8+1+0+0+8+1+0+0+8) so we
get 27000 + 2700 + 270 + 27).

So I think my above list of terms are correct.  Differences from your
list, Diana, are that I have 333 and 1111 and 2222 and 2331 and 2553
and 2775 and 3333 which are missing from your list.  (2331 for example
comes from 399 -> 399 + 939 + 993, and any list of identical digits
maps to itself, so 333 -> 333.)

Diana, would you verify that you agree with my list of terms and then
if you do please submit the corrected terms for this sequence?

Thanks,
--Joshua Zucker


On 7/11/07, Diana Mecum <diana.mecum at gmail.com> wrote:
> Sequence Fans,
>
> I am looking at sequence A071267.
>
> %I A071267
> %S A071267
> 2,4,6,8,10,11,12,14,16,18,22,33,44,55,66,77,88,99,101,110,121,132,143,
> %T A071267 154,165,176,187,198,111,222,666,888,1110
> %N A071267 Numbers which can be expressed as the sum of all distinct digit
> permutations of
>                some number k.
> %C A071267 222 can be expressed so in two different ways i.e. 222= 200 +020
> + 002 as well as
>                222= 101 +110 +011. Question: find a number which can be so
> expressed
>                 in n different ways.
> %e A071267 1110 is a member as a sum of all distinct permutations of 104.
> i.e. 104,140,410,
>                401,014,041.
> %Y A071267 Sequence in context: A081472 A097660 A067030 this_sequence
> A072427 A050420 A096922
> %Y A071267 Adjacent sequences: A071264 A071265 A071266 this_sequence A071268
> A071269 A071270
> %K A071267 base,more,nonn
> %O A071267 1,1
> %A A071267 Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Jun 01 2002
>
> When I try to follow the rule for generating the sequence numbers, I get the
> following list;
>
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 111,
> 121, 132, 143, 154, 165,
>  176, 187, 222, 444, 555, 666, 777, 888, 999, 1110, 1221, 1332, 1443, 1554,
> 1665, 1776, 1887,
>  1998, 2109, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996
>
> Can someone explain why my list differs from the original? I am not
> understanding the hypothesis to generate the original list.
>
> Diana M.
>
> --
> "God made the integers, all else is the work of man."
> L. Kronecker, Jahresber. DMV 2, S. 19.