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A062790 Moebius transform of the cototient function A051953.

%I A062790 #28 Dec 15 2023 06:19:12
%S A062790 0,1,1,1,1,2,1,2,2,4,1,3,1,6,5,4,1,6,1,5,7,10,1,6,4,12,6,7,1,8,1,8,11,
%T A062790 16,9,8,1,18,13,10,1,12,1,11,12,22,1,12,6,20,17,13,1,18,13,14,19,28,1,
%U A062790 13,1,30,16,16,15,20,1,17,23,24,1,16,1,36,24,19,15,24,1,20,18,40,1,19
%N A062790 Moebius transform of the cototient function A051953.
%H A062790 Antti Karttunen, <a href="/A062790/b062790.txt">Table of n, a(n) for n = 1..16384</a> (terms 1 .. 2000 from Harry J. Smith)
%H A062790 Antti Karttunen, <a href="/A062790/a062790.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%H A062790 N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)
%F A062790 a(n) = Sum f(n/d)*mu(d), where d divides n and f(x) = x-phi(x) = A051953(x).
%F A062790 a(n) = A056239(A318836(n)). - _Antti Karttunen_, Nov 24 2018
%F A062790 From _Amiram Eldar_, Dec 15 2023: (Start)
%F A062790 a(n) = A000010(n) - A007431(n).
%F A062790 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 6/Pi^2 - 36/Pi^4. (End)
%e A062790 n = 255, its divisors are {1,3,5,25,17,51,85,255}, A051953(255/d) = {127,21,19,1,7,1,1,0}, mu(d) = {1,-1,-1,1,-1,1,1,-1}, the sum is a(255) = 127-21-19+1-7+1+1+0 = 130-47 = 83.
%t A062790 Table[DirichletConvolve[MoebiusMu[n], n-EulerPhi[n], n, k], {k, 100}]  (* _Amiram Eldar_, Nov 24 2018 *)
%o A062790 (PARI) A062790(n)={
%o A062790     local(a=0) ;
%o A062790     fordiv(n,d,
%o A062790         a += moebius(d)*(n/d-eulerphi(n/d)) ;
%o A062790     ) ;
%o A062790     return(a) ;
%o A062790 } \\ _R. J. Mathar_, Mar 24 2012
%o A062790 (PARI) A062790(n) = sumdiv(n,d,moebius(n/d)*(d-eulerphi(d))); \\ _Antti Karttunen_, Nov 24 2018
%Y A062790 Cf. A000010, A001065, A007431, A051953, A056239, A318836.
%K A062790 nonn
%O A062790 1,6
%A A062790 _Labos Elemer_, Jul 19 2001
%E A062790 OFFSET changed from 0 to 1 by _Harry J. Smith_, Aug 11 2009