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%I #16 Oct 21 2021 22:40:35
%S 0,0,1,1,2,3,4,5,7,7,10,10,14,14,17,17,22,20,28,25,29,30,38,32,43,40,
%T 45,43,57,45,62,56,62,63,70,61,84,74,81,74,98,78,108,92,95,102,120,95,
%U 127,109,123,116,149,118,142,129,145,147,173,126,182,163,164,164,184,158,211
%N Number of partitions of n into 3 parts where at least one of the parts divides the product of the other two.
%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( c(i*(n-i-j)/j) + c(j*(n-i-j)/i) + c(i*j/(n-i-j)) ), where c(n) = 1 - ceiling(n) + floor(n).
%e a(9) = 7; All of the partitions of 9 (into 3 such parts) satisfy these conditions. They are (1,1,7), (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4) and (3,3,3).
%e a(10) = 7; The partitions of 10 into 3 such parts are (1,1,8), (1,2,7), (1,3,6), (1,4,5), (2,2,6), (2,4,4) and (3,3,4).
%t Block[{c}, c[n_] := 1 - Ceiling[n] + Floor[n]; Array[Sum[Sum[Sign[c[i*(# - i - j)/j] + c[j*(# - i - j)/i] + c[i*j/(# - i - j)]], {i, j, Floor[(# - j)/2]}], {j, Floor[#/3]} ] &, 67]] (* _Michael De Vlieger_, Oct 21 2021 *)
%K nonn
%O 1,5
%A _Wesley Ivan Hurt_, Oct 21 2021