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%I #4 Apr 11 2021 23:40:18
%S 0,0,0,0,0,1,0,0,0,2,0,3,0,3,4,2,0,4,0,6,6,5,0,10,0,6,3,9,0,23,0,8,10,
%T 8,12,13,0,9,12,20,0,34,0,15,18,11,0,38,1,14,16,18,0,28,20,30,18,14,0,
%U 61,0,15,27,26,24,56,0,24,22,65,0,43,0,18,30,27,30,67,0,74
%N Number of partitions of n into 3 distinct parts x,y,z such that (x+y+z) | x*y*z.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (1 - ceiling(i*j*(n-i-j)/n) + floor(i*j*(n-i-j)/n)) * (1 - [i = j]) * (1 - [n-i = 2*j]) * (1 - [n-j = 2*i]), where [ ] is the Iverson bracket.
%e a(10) = 2; [1,4,5], [2,3,5], with all parts distinct;
%e a(12) = 3; [1,3,8], [2,4,6], [3,4,5], with all parts distinct.
%t Table[Sum[Sum[(1 - KroneckerDelta[i, j]) (1 - KroneckerDelta[n - j, 2 i]) (1 - KroneckerDelta[n - i, 2 j]) (1 - Ceiling[i*j*(n - i - j)/n] + Floor[i*j*(n - i - j)/n]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 100}]
%Y Cf. A343270.
%K nonn
%O 1,10
%A _Wesley Ivan Hurt_, Apr 11 2021
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