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A168036 Difference between n' and n, where n' is the arithmetic derivative of n (A003415).

%I A168036 #24 Dec 08 2023 12:29:06
%S A168036 0,-1,-1,-2,0,-4,-1,-6,4,-3,-3,-10,4,-12,-5,-7,16,-16,3,-18,4,-11,-9,
%T A168036 -22,20,-15,-11,0,4,-28,1,-30,48,-19,-15,-23,24,-36,-17,-23,28,-40,-1,
%U A168036 -42,4,-6,-21,-46,64,-35,-5,-31,4,-52,27,-39,36,-35,-27,-58,32,-60,-29
%N A168036 Difference between n' and n, where n' is the arithmetic derivative of n (A003415).
%C A168036 Let k = n'-n. For k = -1 n is a primary pseudoperfect number (A054377), apart from n=1; For k=0 n is p^p, being p a prime number (A051674); For k = 1 n is a Giuga number (A007850).
%H A168036 T. D. Noe, <a href="/A168036/b168036.txt">Table of n, a(n) for n = 0..10000</a>
%F A168036 a(A083347(n)) < 0; a(A051674(n)) = 0; a(A083348(n)) > 0. - _Reinhard Zumkeller_, May 22 2015
%F A168036 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = -1 + Sum_{p prime} 1/(p*(p-1)) = A136141 - 1 = -0.226843... . - _Amiram Eldar_, Dec 08 2023
%p A168036 with(numtheory);
%p A168036 A168036:=proc(q)
%p A168036 local n,p;
%p A168036 for n from 0 to q do
%p A168036   print(n*add(op(2,p)/op(1,p),p=ifactors(n)[2])-n); od; end:
%p A168036 A168036(1000); # _Paolo P. Lava_, Nov 05 2012
%t A168036 np[k_] := Module[{f, n, m, p}, If[k < 2, np[k] = 0; Return[0], If[PrimeQ[k], np[k] = 1; Return[1], f = FactorInteger[k, 2]; m = f[[1, 1]]; n = k/m; p = m np[n] + n np[m]; np[k] = p; Return[p]]]];
%t A168036 Table[np[n] - n, {n, 0, 100}] (* _Robert Price_, Mar 14 2020 *)
%o A168036 (Haskell)
%o A168036 a168036 n = a003415 n - n  -- _Reinhard Zumkeller_, May 22 2015
%Y A168036 Cf. A007850, A051674, A054377, A136141.
%Y A168036 Cf. A003415, A051674, A083347, A083348.
%K A168036 easy,sign
%O A168036 0,4
%A A168036 _Paolo P. Lava_, Nov 17 2009