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%I #6 Oct 31 2022 12:12:15
%S 0,0,2,5,7,10,17,22,23,25,36,40,52,59,63,76,84,87,106,112,116,132,155,
%T 159,173,176,191,187,215,208,254,271,261,285,291,301,337,356,364,378,
%U 418,407,471,483,486,513,560,564,591,575,585,609,661,664,672,688,698,738,797,776
%N Total sum of the distinct prime factors of s*t, for all positive integer pairs (s,t) such that s + t = n.
%F a(n) = Sum_{k=1..floor(n/2)} sopf(k*(n-k)).
%e a(7) = 17; The partitions of 7 into two positive integer parts are (6,1), (5,2), and (4,3). The sum of the distinct prime factors of 6*1, 5*2, and 4*3 are 5, 7, and 5 respectively. The total is then 5 + 7 + 5 = 17.
%t Table[Sum[Sum[k (PrimePi[k] - PrimePi[k - 1]) (1 - Ceiling[i (n - i)/k] + Floor[i (n - i)/k]), {k, i (n - i)}], {i, Floor[n/2]}], {n, 80}]
%o (PARI) a(n) = sum(k=1, n\2, vecsum(factor(k*(n-k))[,1])); \\ _Michel Marcus_, Oct 31 2022
%Y Cf. A008472 (sopf).
%K nonn
%O 1,3
%A _Wesley Ivan Hurt_, Jun 08 2021
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