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A300244 Difference between A005187 and its Möbius transform (A297111).

%I A300244 #12 Mar 14 2018 03:50:33
%S A300244 0,1,1,3,1,6,1,7,4,10,1,14,1,13,11,15,1,22,1,22,14,21,1,30,8,25,16,29,
%T A300244 1,40,1,31,22,34,18,46,1,37,26,46,1,57,1,45,38,44,1,62,11,57,35,53,1,
%U A300244 68,26,61,38,56,1,84,1,59,51,63,30,90,1,70,45,89,1,94,1,73,65,77,29,104,1,94,50,81,1,117,39,84,57,93,1,128,33,92,60,91,42
%N A300244 Difference between A005187 and its Möbius transform (A297111).
%H A300244 Antti Karttunen, <a href="/A300244/b300244.txt">Table of n, a(n) for n = 1..16384</a>
%F A300244 a(n) = A005187(n) - A297111(n).
%F A300244 a(n) = -Sum_{d|n, d<n} A008683(n/d)*A005187(d).
%t A300244 Table[IntegerExponent[(2 n)!, 2] - DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 95}] (* or *)
%t A300244 Fold[Function[{a, n}, Append[a, {Abs@ Total@ Map[MoebiusMu[n/#] a[[#, -1]] &, Most@ Divisors@ n], IntegerExponent[(2 n)!, 2]}]], {{0, 1}}, Range[2, 95]][[All, 1]] (* _Michael De Vlieger_, Mar 10 2018 *)
%o A300244 (PARI)
%o A300244 A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
%o A300244 A300244(n) = -sumdiv(n,d,(d<n)*moebius(n/d)*A005187(d));
%Y A300244 Cf. A005187, A008683, A297111, A297114, A297117.
%K A300244 nonn
%O A300244 1,4
%A A300244 _Antti Karttunen_, Mar 10 2018