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

[Thomas Scheuerle]

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