A350032 -id:A350032 - cp's OEIS frontend

A350029 Write n as n = k1 + k2 + ... + km, so that all k are distinct positive integers. a(n) is the maximum value of A001055(k1) + A001055(k2) + ... + A001055(km) over all such partitions.

1, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25

Offset: 1

Thomas Scheuerle, Dec 09 2021

nonn

There exist cases where a(n) < a(n-1). Some examples are n = 53, 77, 113, 125, ...
There may exist multiple partitions of n = k1 + k2 + ... + km, where a(n) = A001055(k1) + A001055(k2) + ... + A001055(km). The number of such partitions is A350032(n).
It appears that a(n) - log(A066739(n)) > 0.
If the definition of this sequence would allow k1 = k2 = km, then this sequence would be the trivial sequence a(n) = n instead.

			  n = k1+k2+...+km   A001055(k1)+...+A001055(km)  = a(n)
--------------------------------------------------------
  1 = 1              1                            = 1
  2 = 2              1                            = 1
  3 = 1 + 2          1 + 1                        = 2
  4 = 1 + 3          1 + 1                        = 2
  5 = 1 + 4          1 + 2                        = 3
  6 = 1 + 2 + 3      1 + 1 + 1                    = 3

Thomas Scheuerle, Examples for n = 1 to 100.

Cf. A000009, A001055, A066739, A350032.

A350029(n, K=0) = { my(a=A001055(n)); while(n>2*K+=1, a=max(A001055(K)+A350029(n-K,K), a) ); a } \\ M. F. Hasler, Dec 09 2021

cp's OEIS Frontend

A350029 Write n as n = k1 + k2 + ... + km, so that all k are distinct positive integers. a(n) is the maximum value of A001055(k1) + A001055(k2) + ... + A001055(km) over all such partitions.

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