A337943 - OEIS

%I #10 Feb 01 2021 20:31:05

%S 0,2,1,5,2,10,4,14,8,14,7,29,9,26,13,33,12,47,16,39,24,43,19,76,20,49,

%T 34,72,24,78,28,82,43,67,31,128,32,83,51,98,37,140,40,122,59,98,44,

%U 189,55,103,70,133,50,185,51,178,85,133,58,217,60,143,107,190,58,235,67,187

%N Total number of divisors, d, of the sum of the squared parts in each partition of n into two parts, where d < n.

%F a(n) = Sum_{i=1..floor(n/2)} Sum_{k=1..n} (1 - ceiling((i^2+(n-i)^2)/k) + floor((i^2+(n-i)^2)/k)).

%e a(4) = 5; 4 has two partitions into two parts (3,1) and (2,2). The sums of the squares of these partitions are 3^2 + 1^2 = 10 and 2^2 + 2^2 = 8. Among the positive integers from 1..4, 1 divides 8 and 10, 2 divides 8 and 10 and 4 divides 8, so a(4) = 5.

%t Table[Sum[Sum[1 - Ceiling[(i^2 + (n - i)^2)/k] + Floor[(i^2 + (n - i)^2)/k], {k, n}], {i, Floor[n/2]}], {n, 100}]

%Y Cf. A337944.

%K nonn

%O 1,2

%A _Wesley Ivan Hurt_, Oct 01 2020