A348535 - OEIS

%I #13 Dec 24 2024 07:34:41

%S 0,0,3,2,7,9,16,18,30,28,49,44,71,69,93,96,135,124,177,146,195,205,

%T 279,242,297,312,362,332,469,383,539,492,557,594,627,570,794,761,834,

%U 758,1022,836,1137,984,1066,1171,1412,1188,1409,1330,1499,1454,1839,1555,1749,1622

%N Sum of all the parts in the partitions of n into 3 parts that divide the product of the other two parts within each partition.

%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)} j * c(i*(n-i-j)/j) + i * c(j*(n-i-j)/i) + (n-i-j) * c(i*j/(n-i-j)), where c(n) = 1 - ceiling(n) + floor(n).

%e a(9) = 30; The partitions of 9 into 3 parts are (1,1,7), (1,2,6), (1,3,5), (1,4,4), (2,2,5), (2,3,4) and (3,3,3). The sums of the parts that divide the product of the other two, within each partition in the list above are, respectively: (1+1) + (1+2) + (1) + (1+4+4) + (2+2) + (2) + (3+3+3) which equals 30.

%t Block[{c}, c[n_] := 1 - Ceiling[n] + Floor[n]; Array[Sum[Sum[j*c[i*(# - i - j)/j] + i*c[j*(# - i - j)/i] + (# - i - j)*c[i*j/(# - i - j)], {i, j, Floor[(# - j)/2]}], {j, Floor[#/3]}] &, 56] ] (* _Michael De Vlieger_, Oct 21 2021 *)

%Y Cf. A348531.

%K nonn

%O 1,3

%A _Wesley Ivan Hurt_, Oct 21 2021