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%I #69 Jun 15 2020 23:21:07
%S 1,1,1,1,2,1,1,1,2,1,1,2,3,1,1,1,2,3,2,1,1,1,1,1,4,1,3,2,1,2,4,1,1,1,
%T 1,2,3,1,5,1,3,2,1,1,1,5,1,2,1,4,1,1,1,1,6,3,1,2,1,1,1,2,4,1,1,6,1,1,
%U 2,2,3,1,1,1,4,7,1,5,1,1,2,1,3,1,2,7,1,1,1,1,2,5,4
%N Number of partitions of k_n into two parts (s,t) such that k_n | s*t, where k_n is the n-th nonsquarefree number (A013929).
%C a(n) >= 1.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%e a(4) = 1; The 4th nonsquarefree number, A013929(4) = 12 has 6 partitions into two parts: (11,1), (10,2), (9,3), (8,4), (7,5) and (6,6) with corresponding products 11, 20, 27, 32, 35, 36. A013929(4) = 12 only divides the product 36, so a(4) = 1.
%e a(5) = 2; The 5th nonsquarefree number, A013929(5) = 16 has 8 partitions into two parts: (15,1), (14,2), (13,3), (12,4), (11,5), (10,6), (9,7) and (8,8) with corresponding products 15, 28, 39, 48, 55, 60, 63 and 64. A013929(5) = 16 divides two of these products, 48 and 64, so a(5) = 2.
%t Table[If[Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[n/2]}] > 0, Sum[(1 - Ceiling[(i*(n - i))/n] + Floor[(i*(n - i))/n]), {i, Floor[n/2]}], {}], {n, 300}] // Flatten
%Y Cf. A013929.
%K nonn
%O 1,5
%A _Wesley Ivan Hurt_, Jun 09 2020
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