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%I #33 Mar 17 2024 03:19:15
%S 0,0,0,1,0,-1,0,5,2,-1,0,0,0,-1,-1,17,0,0,0,2,-1,-1,0,8,4,-1,14,4,0,
%T -11,0,49,-1,-1,-1,5,0,-1,-1,18,0,-13,0,8,6,-1,0,36,6,2,-1,10,0,15,-1,
%U 28,-1,-1,0,-16,0,-1,10,129,-1,-17,0,14,-1,-15,0,33
%N Difference between the arithmetic derivative of n and the sum of proper divisors of n.
%C Observations: at least the first 50 indices of nonnegative terms are also the first 50 terms of A212165. Also at least the first 28 indices of negative terms are also the first 28 terms of A212168, since A212168 is the complement of A212165.
%H Antti Karttunen, <a href="/A211991/b211991.txt">Table of n, a(n) for n = 1..65537</a>
%F a(n) = A003415(n) - A001065(n).
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (A136141 - A013661 + 1) / 2 = 0.0641113... . - _Amiram Eldar_, Mar 17 2024
%t dn[0] = 0; dn[1] = 0; dn[n_?Negative] := -dn[-n]; dn[n_] := Module[{f = Transpose[FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Table[dn[n] - (DivisorSigma[1, n] - n), {n, 100}] (* _T. D. Noe_, Dec 27 2012 *)
%o (PARI)
%o A003415(n) = {my(fac); if(n<1, 0, fac=factor(n); sum(i=1, matsize(fac)[1], n*fac[i, 2]/fac[i, 1]))}; \\ From A003415
%o A211991(n) = (A003415(n) - (sigma(n)-n)); \\ _Antti Karttunen_, Mar 08 2018
%Y Cf. A000203, A001065, A003415.
%Y Cf. A013661, A136141.
%K sign
%O 1,8
%A _Omar E. Pol_, Dec 18 2012