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a(n) is the least k such that n is the number of halving partitions of k (=A349552(k)).
2

%I #25 Jan 03 2025 03:00:55

%S 0,3,5,15,11,19,21,27,37,69,45,43,191,99,75,83,87,85,153,107,157,151,

%T 149,155,183,179,205,173,219,171,213,335,315,307,395,301,309,333,299,

%U 331,339,365,343,469,347,341,429,589,627,587,427,595,659,669,795,599,915,597,603,661,679,619,667,723,691,813,731,877,1181,693,685,811,1253

%N a(n) is the least k such that n is the number of halving partitions of k (=A349552(k)).

%C For m >= 1, let S(m) = {f(m/2), c(m/2)}, where f = floor and c = ceiling. A halving partition of n is a partition p(1) + p(2) + ... + p(k) = n such that p(1) is in S(n) and p(i) is in S(p(i-1)) for i = 2, 3, ..., k.

%e Let f = floor and c = ceiling.

%e a(1) = 0 corresponds to the empty halving partition of 0.

%e a(3) = 5, since 5 is the smallest number with 3 halving partitions:

%e c(5/2) + c(3/2) = 5;

%e c(5/2) + f(3/2) + c(1/2) = 3 + 1 + 1 = 5;

%e f(5/2) + (2/2) + c(1/2) + c(1/2) = 2 + 1 + 1 + 1.

%Y Cf. A349552.

%K nonn

%O 1,2

%A _Clark Kimberling_, Dec 26 2021

%E Corrected and extended by _Max Alekseyev_, Sep 30 2024