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%I #20 Dec 08 2023 12:31:26
%S -1,0,-1,3,-3,5,-5,11,1,5,-9,20,-11,5,2,31,-15,24,-17,26,0,5,-21,56,
%T -9,5,13,32,-27,43,-29,79,-4,5,-10,79,-35,5,-6,78,-39,53,-41,44,27,5,
%U -45,140,-27,38,-10,50,-51,93,-22,100,-12,5,-57,140,-59,5,29,191,-28,73,-65,62,-16,63,-69,207,-71,5,29,68
%N a(n) = sigma(n) + n' - 2n, where n' is the arithmetic derivative of n (A003415) and sigma is the sum of divisors (A000203).
%C Coincides with A003415 only on perfect numbers (A000396).
%F a(n) = A003415(n) - A033879(n) = A000203(n) + A003415(n) - 2*n.
%F a(n) = A001065(n) + A168036(n).
%F a(n) = A344999(n) / A048250(n) = A345049(n) / A173557(n).
%F Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A013661 + A136141 - 2 = 0.418090735898... . - _Amiram Eldar_, Dec 08 2023
%t A003415[n_] := If[n < 2, 0, Module[{f = FactorInteger[n]}, If[PrimeQ[n], 1, Total[n*f[[All, 2]]/f[[All, 1]]]]]];
%t a[n_] := DivisorSigma[1, n] + A003415[n] - 2 n;
%t Array[a, 80] (* _Jean-François Alcover_, Jun 12 2021 *)
%o (PARI)
%o A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A345001(n) = (sigma(n)+A003415(n)-(2*n));
%Y Cf. A000203, A000396, A001065, A003415, A033879, A168036, A344999, A345002, A345003, A345004, A345005, A345049.
%Y Cf. also A051709, A344753 (analogous sequences).
%Y Cf. A013661, A136141.
%K sign,easy
%O 1,4
%A _Antti Karttunen_, Jun 05 2021