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%I #47 Dec 01 2023 15:53:50
%S -1,0,0,0,0,2,0,1,0,2,0,3,0,2,2,1,0,3,0,3,2,2,0,5,0,2,1,3,0,6,0,2,2,2,
%T 2,4,0,2,2,5,0,6,0,3,3,2,0,6,0,3,2,3,0,5,2,5,2,2,0,9,0,2,3,2,2,6,0,3,
%U 2,6,0,7,0,2,3,3,2,6,0,6,1,2,0,9,2,2,2,5,0,9,2,3,2,2,2,8,0,3,3,4,0,6,0,5,6
%N Difference between the number of proper divisors of n and the number of squares dividing n.
%C The difference between the number of ways of writing n = m + k and the number of ways of writing n = r*s, where m|k and r|s.
%C First occurrence of k beginning with k=-1: 1, 2, 8, 6, 12, 36, 24, 30, 72, 96, 60, 2097152, 216, 576, 120, 210, 1152, 240, 864, etc. - _Robert G. Wilson v_, Nov 28 2017
%H Antti Karttunen, <a href="/A293575/b293575.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A032741(n) - A046951(n).
%F a(n) = A056595(n) - 1. - _Antti Karttunen_, Oct 30 2017
%F a(n) = 0 iff n is a prime or a square of a prime, A000430. - _Robert G. Wilson v_, Nov 28 2017
%F Sum_{k=1..n} a(k) ~ n*log(n) + (2*gamma - zeta(2) - 2)*n, where gamma is Euler's constant (A001620). - _Amiram Eldar_, Dec 01 2023
%e a(6) = 2 because 2 is difference of number of ways of writing n = 1 + 5 = 2 + 4 = 3 + 3 where 1|5, 2|4, 3|3 and number of ways of writing n = 1*6 where 1|6.
%t f[n_] := Block[{d = Divisors@ n}, Length@ d - Length[ Select[ d, IntegerQ@ Sqrt@# &]] - 1];; Array[f, 105] (* _Robert G. Wilson v_, Nov 28 2017 *)
%Y One less than A056595.
%Y Cf. A000430, A032741, A046951, A166684, A080257.
%Y Cf. A001620, A013661.
%K sign
%O 1,6
%A _Juri-Stepan Gerasimov_, Oct 14 2017
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