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%I #6 Apr 22 2021 21:53:48
%S 0,0,0,0,0,0,0,0,0,1,0,2,0,2,2,4,0,7,0,9,7,6,0,17,5,8,8,18,0,28,0,20,
%T 13,13,16,44,0,13,18,46,0,50,0,36,37,18,0,77,14,45,24,47,0,71,34,78,
%U 31,24,0,134,0,24,65,74,38,96,0,68,37,108,0,174,0,31,80,76,51
%N Number of partitions of n into 3 parts [r,s,t] such that r < s < t where no part divides any larger part and at least 1 part divides n.
%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)} sign(3 - c(n/j) - c(n/i) - c(n/(n-i-j))) * c(i/j) * c((n-i-j)/i) * c((n-i-j)/j), where c(n) = ceiling(n) - floor(n).
%e a(12) = 2; [2,3,7], [3,4,5] (Not counted [1,1,10], [1,2,9], [1,3,8], [1,4,7], [1,5,6], [2,2,8], [2,4,6], [2,5,5], [3,3,6], [4,4,4]).
%t Table[Sum[Sum[Sign[3 - Ceiling[n/j] + Floor[n/j] - Ceiling[n/(n - i - j)] + Floor[n/(n - i - j)] - Ceiling[n/i] + Floor[n/i]] (Ceiling[i/j] - Floor[i/j]) (Ceiling[(n - i - j)/j] - Floor[(n - i - j)/j]) (Ceiling[(n - i - j)/i] - Floor[(n - i - j)/i]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 100}]
%o (PARI) c(n) = ceil(n)-floor(n)
%o a(n) = sum(j=1, floor(n/3), sum(i=j, floor((n-j)/2), sign(3-c(n/j)-c(n/i)-c(n/(n-i-j))) * c(i/j) * c((n-i-j)/i) * c((n-i-j)/j))) \\ _Felix Fröhlich_, Apr 21 2021
%Y Cf. A343246.
%K nonn
%O 1,12
%A _Wesley Ivan Hurt_, Apr 13 2021
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