Question related to sequence A071267

[Giovanni Resta]

Diana Mecum wrote:
> %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.
> 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.
First of all, you must bear in mind that a consistent fraction of the
sequences submitted by Amarnath Murthy
are erroneous.

In any case it seems to me that also your list is not consistent with
the definition.
In particular, considering elements less or equal to 3996, you list
lacks the following
terms: 333, 1111, 2222, 2331, 2553, 2775, 3333
It is clear that all the repdigits (like 333, 1111, 2222, 3333, and so
on) belong to the
sequence since, for example, the set of all the distinct permutations of
333 is just 333.
(moreover 333 is also produced by the permutations of 300: 300+030+003 =
333).
The other missing values are produced as :
2331 = 399 + 993 + 939
2553 = 599+995+959
2775 = 799+997+979

So I think that the correct list, up to 10000, is
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

while, if we let drop the 'distinct' clause (so for example 11 produces
not 11 but 11+11=22), the
elements  up to 10000 are
1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121,
132,
143, 154, 165, 176, 187, 198, 222, 444, 666, 888, 1110, 1332, 1554, 1776,
1998, 2220, 2442, 2664, 2886, 3108, 3330, 3552, 3774, 3996, 4218, 4440,
4662,
4884, 5106, 5328, 5550, 5772, 5994, 6666

g.