%I
%S 0,3,5,7,8,10,11,10,11,16,16,18,20,15,14,23,23,25,26,17,21,34,30,17,
%T 23,22,18,38,37,39,39,22,31,29,20,45,56,35,25,50,51,53,56,24,34,70,56,
%U 23,24,30,35,68,62,25,27,33,51,88,69,71,92,44,23,28,32,81,86,45,38,83,81,83,110,50,34,39,34,95,90
%N Sum of primes dividing nth triangular number, counted with multiplicity.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SumofPrimeFactors.html">Sum of prime factors</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TriangularNumber.html">Triangular number</a>
%F For any integer coefficient C(n) of the polynomial generated by the Triangular Numbers generating function f(x)=x/((1x)^3), if C(n) = Product (p_j^k_j) then a(n) = Sum (p_j * k_j).
%F a(n) = A001414(A000217(n)).
%e a(4) = 7; the 4th triangular number is 10, the prime factors of 10 are 2 and 5, and 2+5 = 7.
%e a(6) = 10; the 6th triangular number is 21, the prime factors of 21 are 3 and 7, and 3+7 = 10.
%t a[1]=0; a[n_] := Plus @@ Times @@@ FactorInteger[n (n+1)/2]; Array[a, 80] (* _Giovanni Resta_, Jun 12 2016 *)
%o (PARI) a(n) = my(f=factor(n*(n+1)/2));sum(i=1,matsize(f)[1],f[i,1]*f[i,2]) \\ _David A. Corneth_, Jun 12 2016
%Y Cf. A000217 (triangular numbers), A001414 (sum of primes dividing n).
%K nonn
%O 1,2
%A _Luca Pezzullo_, Jun 11 2016
%E a(30) and a(38) corrected by _Giovanni Resta_, Jun 12 2016
