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A364808 a(n) = sum of minimal runlengths of all the partitions of n.

%I A364808 #17 Sep 17 2023 20:12:39
%S A364808 1,3,5,9,11,20,22,36,44,63,74,114,128,180,224,298,355,485,573,760,922,
%T A364808 1174,1419,1836,2189,2756,3341,4160,4988,6217,7412,9131,10941,13326,
%U A364808 15916,19379,22988,27770,33017,39662,46919,56223,66308,79047,93187,110512
%N A364808 a(n) = sum of minimal runlengths of all the partitions of n.
%H A364808 Alois P. Heinz, <a href="/A364808/b364808.txt">Table of n, a(n) for n = 1..2000</a>
%e A364808 The partitions of 4 are [4], [3,1], [2,2], [2,1,1], [1,1,1,1], with runlengths {1}, {1,1}, {2}, {1,2}, {4} having minima 1, 1, 2, 1, 4, with sum 9, so that a(4) = 9.
%p A364808 b:= proc(n, i, m) option remember; `if`(n=0, m, `if`(i=1, min(m, n),
%p A364808       add(b(n-i*j, i-1, `if`(j=0, m, min(m, j))), j=0..n/i)))
%p A364808     end:
%p A364808 a:= n-> b(n$3):
%p A364808 seq(a(n), n=1..50);  # _Alois P. Heinz_, Sep 17 2023
%t A364808 m[n_] := m[n] = Map[Split, IntegerPartitions[n]]
%t A364808 t[n_] := t[n] = Table[Map[Length, m[n][[k]]], {k, 1, PartitionsP[n]}]
%t A364808 Table[Total[Map[Min, t[n]]], {n, 1, 47}]
%o A364808 (Python)
%o A364808 from sympy.utilities.iterables import partitions
%o A364808 def A364808(n): return sum(min(p.values()) for p in partitions(n)) # _Chai Wah Wu_, Sep 17 2023
%Y A364808 Cf. A000041, A264397 (sum of maximal runlengths).
%K A364808 nonn
%O A364808 1,2
%A A364808 _Clark Kimberling_, Sep 10 2023