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

0,5

Links

David A. Corneth, Table of n, a(n) for n = 0..10000

Index entries for sequences related to partitions

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).

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A072670, A183003, A350534.

Sequence in context: A238661 A135525 A319142 * A117538 A001060 A042343

Adjacent sequences: A350494 A350495 A350496 * A350498 A350499 A350500

Keyword

nonn

Author

Wesley Ivan Hurt, Jan 03 2022

Extensions

a(0) = 0 prepended by David A. Corneth, Jan 09 2022

Status

approved