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

Author

Wesley Ivan Hurt, Jan 03 2022

Keywords

• nonn

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

• David A. Corneth, Table of n, a(n) for n = 0..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A072670, A183003, A350534.

Programs

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