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A306927 a(n) = A001615(n) - n.

%I A306927 #27 Dec 08 2023 12:29:10
%S A306927 0,1,1,2,1,6,1,4,3,8,1,12,1,10,9,8,1,18,1,16,11,14,1,24,5,16,9,20,1,
%T A306927 42,1,16,15,20,13,36,1,22,17,32,1,54,1,28,27,26,1,48,7,40,21,32,1,54,
%U A306927 17,40,23,32,1,84,1,34,33,32,19,78,1,40,27,74,1,72
%N A306927 a(n) = A001615(n) - n.
%C A306927 Analogous to A051953.
%C A306927 a(n) = A051953(n) if n is an element of A000961.
%C A306927 a(n) > A051953(n) if n is an element of A024619.
%C A306927 The sum of the proper divisors d of n such that n/d is squarefree. - _Amiram Eldar_, Sep 06 2019
%H A306927 Amiram Eldar, <a href="/A306927/b306927.txt">Table of n, a(n) for n = 1..10000</a>
%H A306927 Wikipedia, <a href="http://en.wikipedia.org/wiki/Dedekind_psi_function">Dedekind psi function</a>.
%F A306927 a(n) = A001615(n) - n.
%F A306927 a(n) = Sum_{d|n, d<n} (mu(n/d)^2 * d). - _Amiram Eldar_, Sep 06 2019
%F A306927 Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n*log(n)), where c = 15/Pi^2 - 1 = 0.519817... . - _Amiram Eldar_, Dec 08 2023
%e A306927 0 is a term because A001615(1) - 1 = 0.
%e A306927 1 is a term because A001615(2) - 2 = 1.
%e A306927 3 is a term because A001615(9) - 9 = 3.
%t A306927 a[1] = 0; a[n_] := n * (Times @@ (1 + 1/FactorInteger[n][[;; , 1]]) - 1); Array[a, 100] (* _Amiram Eldar_, Sep 06 2019 *)
%o A306927 (PARI) a(n) = n*(sumdivmult(n, d, issquarefree(d)/d) - 1); \\ _Michel Marcus_, Mar 18 2019
%Y A306927 Cf. A000961, A001615, A024619, A051953, A082020.
%K A306927 nonn,easy
%O A306927 1,4
%A A306927 _Torlach Rush_, Mar 16 2019