This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A068312 #17 May 14 2025 04:27:24 %S A068312 0,0,1,5,7,8,10,32,60,39,16,61,71,20,71,244,212,111,123,143,247,131, %T A068312 34,380,520,155,378,621,275,247,263,1008,1280,271,239,951,795,56,343, %U A068312 1256,1004,431,451,581,1443,942,70,2092,2492,840 %N A068312 Arithmetic derivative of triangular numbers. %H A068312 Reinhard Zumkeller, <a href="/A068312/b068312.txt">Table of n, a(n) for n = 0..10000</a> %F A068312 a(n) = A003415(A000217(n)). %F A068312 For n > 1: a(n) = A258197(n,2) = A258197(n,n-2). - _Reinhard Zumkeller_, May 26 2015 %e A068312 a(7) = d(7*8/2) = d(28) = d(2*14) = d(2)*14 + 2*d(14) = 1*14 + 2*d(2*7) = 14 + 2*(2*d(7) + d(2)*7) = 14 + 2*(2*1 + 1*7) = 14 + 2*9 = 14 + 18 = 32, where d(n) = A003415(n) with d(1) = 0, d(prime) = 1 and d(m*n) = d(m)*n + m*d(n). %t A068312 a[0] = a[1] = 0; a[n_] := (n*(n+1)/2) * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n*(n+1)/2]); Array[a, 100, 0] (* _Amiram Eldar_, May 14 2025 *) %o A068312 (Haskell) %o A068312 a068312 = a003415 . a000217 -- _Reinhard Zumkeller_, May 26 2015 %o A068312 (Python) %o A068312 from sympy import factorint %o A068312 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 A068312 Cf. A000217, A003415, A007318, A258197. %K A068312 nonn,easy %O A068312 0,4 %A A068312 _Reinhard Zumkeller_, Feb 25 2002 %E A068312 a(0)=0 prepended by _Reinhard Zumkeller_, May 26 2015