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A344584 Difference between the inverse Möbius transform of the arithmetic derivative of n and the sum of the proper divisors of n: a(n) = A319684(n) - A001065(n).

%I A344584 #15 Dec 09 2021 01:30:59
%S A344584 0,0,0,2,0,1,0,10,3,1,0,11,0,1,1,34,0,13,0,15,1,1,0,47,5,1,21,19,0,12,
%T A344584 0,98,1,1,1,59,0,1,1,67,0,14,0,27,22,1,0,151,7,21,1,31,0,76,1,87,1,1,
%U A344584 0,82,0,1,28,258,1,18,0,39,1,16,0,203,0,1,26,43,1,20,0,219,102,1,0,104,1,1,1,127,0,99,1,51,1,1,1,423
%N A344584 Difference between the inverse Möbius transform of the arithmetic derivative of n and the sum of the proper divisors of n: a(n) = A319684(n) - A001065(n).
%H A344584 Antti Karttunen, <a href="/A344584/b344584.txt">Table of n, a(n) for n = 1..20000</a>
%F A344584 a(n) = A319684(n) - A001065(n) = A211991(n) + A319683(n).
%F A344584 a(n) = Sum_{d|n} A344178(d).
%F A344584 a(n) = n + Sum_{d|n} A168036(d).
%t A344584 Block[{a}, a[1] = 0; a[n_] := a[n] = If[n < 2, 0, n Total[#2/#1 & @@@ FactorInteger[n]]]; Array[DivisorSum[#, a[#] &] - DivisorSigma[1, #] + # &, 96]] (* _Michael De Vlieger_, May 24 2021 *)
%o A344584 (PARI)
%o A344584 A003415(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]/f[i, 1]));
%o A344584 A319684(n) = sumdiv(n, d, A003415(d));
%o A344584 A344584(n) = (A319684(n) - (sigma(n)-n));
%Y A344584 Cf. A000203, A001065, A003415, A168036, A211991, A319683, A319684.
%Y A344584 Inverse Möbius transform of A344178.
%K A344584 nonn
%O A344584 1,4
%A A344584 _Antti Karttunen_, May 24 2021