A350497 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.
0, 0, 0, 1, 2, 5, 7, 12, 19, 27, 32, 48, 55, 68, 84, 109, 120, 149, 162, 196, 223, 249, 266, 323, 359, 392, 430, 484, 509, 586, 614, 678, 728, 775, 831, 952, 989, 1044, 1106, 1219, 1261, 1379, 1424, 1520, 1627, 1698, 1748, 1919, 2009, 2124, 2213, 2332, 2392, 2552, 2655, 2827
Offset: 0
Keywords
Examples
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.
Links
Programs
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Mathematica
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}] Table[Total[Select[IntegerPartitions[n,{3}],#[[1]]>=Times@@Rest[#]&][[All,1]]],{n,0,60}] (* Harvey P. Dale, Aug 22 2022 *) -
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
Formula
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).
Extensions
a(0) = 0 prepended by David A. Corneth, Jan 09 2022