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 A211991 #33 Mar 17 2024 03:19:15
%S A211991 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 A211991 -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 A211991 28,-1,-1,0,-16,0,-1,10,129,-1,-17,0,14,-1,-15,0,33
%N A211991 Difference between the arithmetic derivative of n and the sum of proper divisors of n.
%C A211991 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 A211991 Antti Karttunen, <a href="/A211991/b211991.txt">Table of n, a(n) for n = 1..65537</a>
%F A211991 a(n) = A003415(n) - A001065(n).
%F A211991 Sum_{k=1..n} a(k) ~ c * n^2, where c = (A136141 - A013661 + 1) / 2 = 0.0641113... . - _Amiram Eldar_, Mar 17 2024
%t A211991 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 A211991 (PARI)
%o A211991 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 A211991(n) = (A003415(n) - (sigma(n)-n)); \\ _Antti Karttunen_, Mar 08 2018
%Y A211991 Cf. A000203, A001065, A003415.
%Y A211991 Cf. A013661, A136141.
%K A211991 sign
%O A211991 1,8
%A A211991 _Omar E. Pol_, Dec 18 2012