|
%I #27 Dec 01 2023 15:53:11
%S 0,1,1,2,1,3,1,2,2,3,1,4,1,3,3,2,1,4,1,4,3,3,1,4,2,3,2,4,1,7,1,2,3,3,
%T 3,4,1,3,3,4,1,7,1,4,4,3,1,4,2,4,3,4,1,4,3,4,3,3,1,8,1,3,4,2,3,7,1,4,
%U 3,7,1,4,1,3,4,4,3,7,1,4,2,3,1,8,3,3,3,4,1,8,3,4,3,3,3,4,1,4,4,4,1,7,1,4,7
%N a(n) is the number of proper divisors of n that are squarefree.
%H Antti Karttunen, <a href="/A293227/b293227.txt">Table of n, a(n) for n = 1..16384</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F a(n) = Sum_{d|n, d<n} A008966(d).
%F a(n) = A034444(n) - A008966(n).
%F a(n) = 2^A001221(n) - A008683(n)^2 = 2^omega(n) - mu(n)^2.
%F G.f.: Sum_{k>=1} mu(k)^2*x^(2*k)/(1 - x^k). - _Ilya Gutkovskiy_, Oct 28 2018
%F Sum_{k=1..n} a(k) ~ (6/Pi^2)*n*(log(n) + 2*(gamma - 1 - zeta'(2)/zeta(2))), where gamma is Euler's constant (A001620). - _Amiram Eldar_, Dec 01 2023
%p with(numtheory): seq(coeff(series(add(mobius(k)^2*x^(2*k)/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 120); # _Muniru A Asiru_, Oct 28 2018
%t Table[Count[Most[Divisors[n]],_?SquareFreeQ],{n,110}] (* _Harvey P. Dale_, Jun 15 2021 *)
%t a[n_] := 2^PrimeNu[n] - Boole[SquareFreeQ[n]]; Array[a, 100] (* _Amiram Eldar_, Jun 30 2022 *)
%o (PARI) A293227(n) = sumdiv(n, d, (d<n)*issquarefree(d));
%Y Cf. A001221, A005117, A008683, A008966, A034444, A293228, A293234.
%Y Cf. A001620, A306016.
%K nonn
%O 1,4
%A _Antti Karttunen_, Oct 08 2017
|
