%I #34 Aug 22 2022 13:59:21
%S 0,0,0,1,2,5,7,12,19,27,32,48,55,68,84,109,120,149,162,196,223,249,
%T 266,323,359,392,430,484,509,586,614,678,728,775,831,952,989,1044,
%U 1106,1219,1261,1379,1424,1520,1627,1698,1748,1919,2009,2124,2213,2332,2392,2552,2655,2827
%N Sum of the largest parts in all the partitions of n into 3 parts whose largest part is greater than or equal to the product of the other two.
%H David A. Corneth, <a href="/A350497/b350497.txt">Table of n, a(n) for n = 0..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((n-i-k)/(i*k))) * (n-i-k).
%e a(7) = 12 since we have 7 = 1+1+5 = 1+2+4 = 1+3+3, and the sum of the largest parts in each partition is 5+4+3 = 12. The partition 2+2+3 is not included since 2*2 > 3.
%t Table[Sum[Sum[(n - i - k) Sign[Floor[(n - i - k)/(i*k)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]
%t Table[Total[Select[IntegerPartitions[n,{3}],#[[1]]>=Times@@Rest[#]&][[All,1]]],{n,0,60}] (* _Harvey P. Dale_, Aug 22 2022 *)
%o (PARI) first(n) = my(res=vector(n, i, [0, 0])); for(i = 1, n\2, for(j = i, n\i, c = i + j + i * j; if(c <= n, res[c][1]++; res[c][2] += i*j))); forstep(i = n, 1, -1, for(j = i + 1, n, res[j][2] += ((j-i) * res[i][1] + res[i][2]))); concat(0, vector(#res, i, res[i][2])) \\ _David A. Corneth_, Jan 07 2022
%Y Cf. A072670, A183003, A350534.
%K nonn
%O 0,5
%A _Wesley Ivan Hurt_, Jan 03 2022
%E a(0) = 0 prepended by _David A. Corneth_, Jan 09 2022