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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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%I #28 Jan 11 2022 21:35:09

%S 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,

%T 12,12,13,13,13,14,14,14,14,15,15,15,16,16,16,17,17,17,18,18,18,19,18,

%U 19,19,19,20,20,20,21,21,21,21,22,22,22,23,23,23,24,24,24,25,25

%N 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.

%C There exist cases where a(n) < a(n-1). Some examples are n = 53, 77, 113, 125, ...

%C 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).

%C It appears that a(n) - log(A066739(n)) > 0.

%C If the definition of this sequence would allow k1 = k2 = km, then this sequence would be the trivial sequence a(n) = n instead.

%H Thomas Scheuerle, <a href="/A350029/a350029.txt">Examples for n = 1 to 100</a>.

%e n = k1+k2+...+km A001055(k1)+...+A001055(km) = a(n)

%e --------------------------------------------------------

%e 1 = 1 1 = 1

%e 2 = 2 1 = 1

%e 3 = 1 + 2 1 + 1 = 2

%e 4 = 1 + 3 1 + 1 = 2

%e 5 = 1 + 4 1 + 2 = 3

%e 6 = 1 + 2 + 3 1 + 1 + 1 = 3

%o (PARI) 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

%Y Cf. A000009, A001055, A066739, A350032.

%K nonn

%O 1,3

%A _Thomas Scheuerle_, Dec 09 2021