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A129283 a(n) = (arithmetic derivative of n) + n.

%I A129283 #18 May 14 2025 04:27:49
%S A129283 0,1,3,4,8,6,11,8,20,15,17,12,28,14,23,23,48,18,39,20,44,31,35,24,68,
%T A129283 35,41,54,60,30,61,32,112,47,53,47,96,38,59,55,108,42,83,44,92,84,71,
%U A129283 48,160,63,95,71,108,54,135,71,148,79,89,60,152,62,95,114,256,83,127,68
%N A129283 a(n) = (arithmetic derivative of n) + n.
%H A129283 Reinhard Zumkeller, <a href="/A129283/b129283.txt">Table of n, a(n) for n = 0..10000</a>
%F A129283 a(n) = A003415(n) + n.
%F A129283 a(n) = A003415(n*A051674(k)) / A051674(k).
%F A129283 a(A129284(n)) > 1, a(A129285(n)) > 1, a(A129286(n)) > 1.
%F A129283 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 + Sum_{p prime} 1/(p*(p-1)) = 1 + A136141 = 1.773156... . - _Amiram Eldar_, May 14 2025
%p A129283 A129283 := proc(n)
%p A129283     n+A003415(n) ;
%p A129283 end proc:
%p A129283 seq(A129283(n),n=0..40) ; # _R. J. Mathar_, Feb 04 2022
%t A129283 ad[n_] := Switch[n, 0|1, 0, _?PrimeQ, 1, _, Sum[Module[{p, e}, {p, e} = pe; n*e/p], {pe, FactorInteger[n]}]];
%t A129283 a[n_] := ad[n] + n;
%t A129283 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, May 23 2023 *)
%o A129283 (Haskell)
%o A129283 a129283 n = a003415 n + n  -- _Reinhard Zumkeller_, Nov 01 2013
%Y A129283 Cf. A003415, A051674, A129284, A129285, A129286, A136141.
%K A129283 nonn,easy
%O A129283 0,3
%A A129283 _Reinhard Zumkeller_, Apr 07 2007