[seqfan] Re: A simple looking sequence is spontaneously breaking its monotony

[Jack Grahl]

Do k1,k2, etc all have to be distinct? Or does it just happen to be the
case in the examples you showed?

On Tue, 26 Oct 2021, 09:09 Thomas Scheuerle via SeqFan, <
seqfan at list.seqfan.eu> wrote:

>
> Hi,
>
> An example how easy looking patterns are dangerous ..
>
> Some day in the future I will try to submit this sequence:
> 1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9 ...
> Without knowing its definition it looks a lot like floor(some function of
> n).
> Its definition:
> Split n into a sum n = k1+k2+..km such that a(n) =
> A001055(k1)+...A001055(km) becomes maximal.
> A001055(km)  is the number of ways of factoring km with all factors
> greater than 1.
>
> There are yet two cases known to me where
> a(n+1) < a(n) this is  at n = 52 and 76.
>
> There may be several sums for each n, which reach a(n), but some examples
> of the lexicographically earliest are found here:
>
>   n = k1+k2+..+km          A001055(k1)+...A001055(km)          = a(n)
> n's for prefixes in the sum
>
> --------------------------------------------------------------------------------------------------------
>   1 = 1                             1                          = 1
>   2 = 2                             1                          = 1
>   3 = 1+2                           1+1                        = 2       1
>   4 = 1+3                           1+1                        = 2       1
>   5 = 1+4                           1+2                        = 3       1
>   6 = 1+2+3                         1+1+1                      = 3
>  1;3
>   7 = 1+2+4                         1+1+1                      = 4
>  1;3
>   8 = 1+3+4                         1+1+2                      = 4
>  1;4
>   9 = 2+3+4                         1+1+2                      = 4       2
>  10 = 1+2+3+4                       1+1+1+2                    = 5
>  1;3;6
>  11 = 1+4+6                         1+2+2                      = 5
>  1;5
>  12 = 1+2+4+5                       1+1+2+1                    = 5
>  1;3;7
>  13 = 1+2+4+6                       1+1+2+2                    = 6
>  1;3;7
>  14 = 1+3+4+6                       1+1+2+2                    = 6
>  1;4;8
>  15 = 1+2+4+8                       1+1+2+3                    = 7
>  1;3;7;
>  16 = 1+2+3+4+6                     1+1+1+2+2                  = 7
>  1;3;6;10
>  17 = 2+3+4+8                       1+1+2+3                    = 7
>  2;9
>  18 = 1+2+3+4+8                     1+1+1+2+3                  = 8
>  1;3;6;10
>  19 = 1+4+6+8                       1+2+2+3                    = 8
>  1;5;11
>  20 = 1+2+4+5+8                     1+1+2+1+3                  = 8
>  1;3;7;12
>  21 = 1+2+4+6+8                     1+1+2+2+3                  = 9
>  1;3;7;13
>  22 = 1+3+4+6+8                     1+1+2+2+3                  = 9
>  1;4;8;14
>  23 = 1+2+3+4+5+8                   1+1+1+2+1+3                = 9
>  1;3;6;10
> ......
>
> kindest regards
>
> Thomas Scheuerle
>
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>