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%I #11 Jun 24 2022 17:16:17
%S 0,0,1,5,7,8,10,32,60,39,16,61,71,20,71,244,212,111,123,143,247,131,
%T 34,380,520,155,378,621,275,247,263,1008,1280,271,239,951,795,56,343,
%U 1256,1004,431,451,581,1443,942,70,2092,2492,840
%N Arithmetic derivative of triangular numbers.
%C For n > 1: a(n) = A258197(n,2) = A258197(n,n-2). - _Reinhard Zumkeller_, May 26 2015
%H Reinhard Zumkeller, <a href="/A068312/b068312.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = A003415(A000217(n)).
%e a(7) = d(7*8/2) = d(28) = d(2*14) = d(2)*14 + 2*d(14) =
%e = 1*14 + 2*d(2*7) = 14 + 2*(2*d(7) + d(2)*7) =
%e = 14 + 2*(2*1 + 1*7) = 14 + 2*9 = 14 + 18 = 32;
%e where d(n) = A003415(n)
%e with d(1)=0, d(prime)=1 and d(m*n)=d(m)*n+m*d(n).
%o (Haskell)
%o a068312 = a003415 . a000217 -- _Reinhard Zumkeller_, May 26 2015
%o (Python)
%o from sympy import factorint
%o def A068312(n): return 0 if n <= 1 else ((n+1)*sum((n*e//p for p,e in factorint(n).items()))+ sum(((n+1)*e//p for p,e in factorint(n+1).items()))*n - (n*(n+1)//2))//2 # _Chai Wah Wu_, Jun 24 2022
%Y Cf. A003415, A007318, A258197.
%K nonn
%O 0,4
%A _Reinhard Zumkeller_, Feb 25 2002
%E a(0)=0 prepended by _Reinhard Zumkeller_, May 26 2015
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