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A300239 Difference between A032742 (the largest proper divisor of n) and its Möbius transform (A300236).

%I A300239 #11 Mar 14 2018 03:50:29
%S A300239 0,1,1,1,1,1,1,2,1,1,1,4,1,1,1,4,1,5,1,6,1,1,1,8,1,1,3,8,1,11,1,8,1,1,
%T A300239 1,12,1,1,1,12,1,15,1,12,7,1,1,16,1,9,1,14,1,15,1,16,1,1,1,22,1,1,9,
%U A300239 16,1,23,1,18,1,17,1,24,1,1,9,20,1,27,1,24,9,1,1,30,1,1,1,24,1,29,1,24,1,1,1,32,1,13,13,30,1,35,1,28,17
%N A300239 Difference between A032742 (the largest proper divisor of n) and its Möbius transform (A300236).
%H A300239 Antti Karttunen, <a href="/A300239/b300239.txt">Table of n, a(n) for n = 1..65537</a>
%F A300239 a(n) = A032742(n) - A300236(n).
%F A300239 a(n) = -Sum_{d|n, d<n} A008683(n/d)*A032742(d).
%t A300239 Table[n/FactorInteger[n][[1, 1]] - DivisorSum[n, # MoebiusMu[n/#]/FactorInteger[#][[1, 1]] &], {n, 105}] (* or *)
%t A300239 Fold[Function[{a, n}, Append[a, {Abs@ Total@ Map[MoebiusMu[n/#] a[[#, -1]] &, Most@ Divisors@ n], n/FactorInteger[n][[1, 1]]}]], {{0, 1}}, Range[2, 105]][[All, 1]] (* _Michael De Vlieger_, Mar 10 2018 *)
%o A300239 (PARI)
%o A300239 A032742(n) = if(1==n,n,n/vecmin(factor(n)[,1]));
%o A300239 A300239(n) = -sumdiv(n,d,(d<n)*moebius(n/d)*A032742(d));
%Y A300239 Cf. A008683, A032742, A300236.
%K A300239 nonn
%O A300239 1,8
%A A300239 _Antti Karttunen_, Mar 10 2018