This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A350029 #28 Jan 11 2022 21:35:09
%S A350029 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 A350029 12,12,13,13,13,14,14,14,14,15,15,15,16,16,16,17,17,17,18,18,18,19,18,
%U A350029 19,19,19,20,20,20,21,21,21,21,22,22,22,23,23,23,24,24,24,25,25
%N 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.
%C A350029 There exist cases where a(n) < a(n-1). Some examples are n = 53, 77, 113, 125, ...
%C A350029 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 A350029 It appears that a(n) - log(A066739(n)) > 0.
%C A350029 If the definition of this sequence would allow k1 = k2 = km, then this sequence would be the trivial sequence a(n) = n instead.
%H A350029 Thomas Scheuerle, <a href="/A350029/a350029.txt">Examples for n = 1 to 100</a>.
%e A350029 n = k1+k2+...+km A001055(k1)+...+A001055(km) = a(n)
%e A350029 --------------------------------------------------------
%e A350029 1 = 1 1 = 1
%e A350029 2 = 2 1 = 1
%e A350029 3 = 1 + 2 1 + 1 = 2
%e A350029 4 = 1 + 3 1 + 1 = 2
%e A350029 5 = 1 + 4 1 + 2 = 3
%e A350029 6 = 1 + 2 + 3 1 + 1 + 1 = 3
%o A350029 (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 A350029 Cf. A000009, A001055, A066739, A350032.
%K A350029 nonn
%O A350029 1,3
%A A350029 _Thomas Scheuerle_, Dec 09 2021