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%I #20 Mar 16 2022 02:49:45
%S 0,0,0,1,0,1,0,5,1,1,0,6,0,1,1,17,0,7,0,8,1,1,0,22,1,1,8,10,0,9,0,49,
%T 1,1,1,28,0,1,1,32,0,11,0,14,10,1,0,66,1,11,1,16,0,35,1,42,1,1,0,40,0,
%U 1,12,129,1,15,0,20,1,13,0,88,0,1,12,22,1,17,0,100,43,1,0,52,1,1,1,62,0,49,1,26,1,1,1
%N a(n) = A003415(n) + A003958(n) - n, where A003415 is the arithmetic derivative and A003958 is fully multiplicative with a(p) = (p-1).
%C No negative terms. See comments in A322582.
%C This is the difference between the arithmetic derivative of n [= A003415(n)] and its guaranteed lower bound A322582(n) [= n - A003958(n)].
%H Antti Karttunen, <a href="/A348975/b348975.txt">Table of n, a(n) for n = 1..16384</a>
%F a(n) = A003415(n) - A322582(n).
%F a(n) = A003958(n) + A168036(n).
%t MapAt[# + 1 &, Array[If[# < 2, 0, # Total[#2/#1 & @@@ #2]] + Times @@ Map[(#1 - 1)^#2 & @@ # &, #2] - #1 & @@ {#, FactorInteger[#]} &, 95], 1] (* _Michael De Vlieger_, Mar 15 2022 *)
%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 A003958(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1]--); factorback(f); };
%o A322582(n) = (n-A003958(n));
%o A348975(n) = (A003415(n) - A322582(n));
%Y Cf. A003415, A003958, A168036, A322582.
%Y Cf. also A348970 for the corresponding difference from a guaranteed upper bound.
%K nonn,look
%O 1,8
%A _Antti Karttunen_, Nov 09 2021