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A360523 a(n) = Sum_{d|n} mu(rad(d)) * delta_d(n/d), where rad(n) = A007947(n) and delta_d(n) is the greatest divisor of n that is relatively prime to d.

%I A360523 #10 Feb 11 2023 07:56:49
%S A360523 1,1,2,2,4,2,6,5,7,4,10,4,12,6,8,12,16,7,18,8,12,10,22,10,23,12,24,12,
%T A360523 28,8,30,27,20,16,24,14,36,18,24,20,40,12,42,20,28,22,46,24,47,23,32,
%U A360523 24,52,24,40,30,36,28,58,16,60,30,42,58,48,20,66,32,44,24
%N A360523 a(n) = Sum_{d|n} mu(rad(d)) * delta_d(n/d), where rad(n) = A007947(n) and delta_d(n) is the greatest divisor of n that is relatively prime to d.
%C A360523 Analogous to the Euler totient function (A000010) as A360522 is analogous to A000203.
%H A360523 Amiram Eldar, <a href="/A360523/b360523.txt">Table of n, a(n) for n = 1..10000</a>
%H A360523 Mizan R. Khan, <a href="http://www.pphmj.com/abstract/699.htm">A variant of the divisor functions sigma_a(n)</a>, JP Journal of Algebra, Number Theory and Applications, Vol. 5, No. 3 (2005), pp. 561-574.
%F A360523 Multiplicative with a(p^e) = p^e - e.
%F A360523 Dirichlet g.f.: zeta(s-1)*zeta(s)^2 * Product_{p prime} (1 - 3/p^s + 1/p^(2*s-1) + 1/p^(2*s)).
%F A360523 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - p/((p-1)*(p+1)^2)) = 0.3243742337... .
%F A360523 A000010(n) <= a(n) <= A047994(n) (Khan, 2005).
%F A360523 a(n) = A000010(n) if and only if n is in A078779 (i.e., n is either squarefree or twice a squarefree number).
%F A360523 a(n) = A047994(n) if and only if n is in A005117 (i.e., n is squarefree).
%t A360523 f[p_, e_] := p^e - e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A360523 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] - f[i,2]);}
%Y A360523 Cf. A000010, A000203, A005117, A007947 (rad), A008683 (mu), A047994, A078779, A360522.
%K A360523 nonn,mult
%O A360523 1,3
%A A360523 _Amiram Eldar_, Feb 10 2023