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A357684 The squarefree part (A007913) of numbers whose squarefree part is a unitary divisor (A335275).

%I A357684 #13 Oct 12 2022 05:22:26
%S A357684 1,2,3,1,5,6,7,1,10,11,3,13,14,15,1,17,2,19,5,21,22,23,1,26,7,29,30,
%T A357684 31,33,34,35,1,37,38,39,41,42,43,11,5,46,47,3,1,2,51,13,53,55,57,58,
%U A357684 59,15,61,62,7,1,65,66,67,17,69,70,71,73,74,3,19,77,78,79
%N A357684 The squarefree part (A007913) of numbers whose squarefree part is a unitary divisor (A335275).
%H A357684 Amiram Eldar, <a href="/A357684/b357684.txt">Table of n, a(n) for n = 1..10000</a>
%H A357684 Rafael Jakimczuk, <a href="https://doi.org/10.12988/imf.2017.7113">Two Topics in Number Theory: Sum of Divisors of the Primorial and Sum of Squarefree Parts</a>, International Mathematical Forum, Vol. 12, No. 7 (2017), pp. 331-338.
%F A357684 a(n) = A007913(A335275(n)).
%F A357684 a(n) = 1 iff A335275(n) is a square (A000290).
%F A357684 a(n) = A335275(n) iff A335275(n) is squarefree (A005117).
%F A357684 Sum_{k, a(k) <= x} ~ c*x^2 + o(x^2), where c = (3/Pi^2) * Sum_{k>=1} f(k)/k^4 = 0.32103327852028541131..., and f(k) = Product_{p prime | k} (p/(p+1)) (Jakimczuk, 2017).
%F A357684 Sum_{k=1..n} a(k) ~ c'*x^2 + o(x^2), where c' = c / (A065465)^2 = 0.41313480468422995583... .
%t A357684 s[n_] := If[AllTrue[(f = FactorInteger[n])[[;; , 2]], # == 1 || EvenQ[#] &], i = Position[f[[;; , 2]], 1] // Flatten; Times @@ f[[i, 1]], Nothing]; Array[s, 100]
%o A357684 (PARI) s(n) = {my(f = factor(n), ans = 1); for(k = 1, #f~, if(f[k,2] > 1 && f[k,2]%2, ans = 0)); if(ans, ans = prod(k = 1, #f~, if(f[k,2] == 1, f[k,1], 1))) };
%o A357684 for(n = 1, 100, if(s(n) > 0, print1(s(n), ", ")))
%Y A357684 Cf. A000290, A005117, A007913, A008833, A065465, A069891, A335275.
%K A357684 nonn
%O A357684 1,2
%A A357684 _Amiram Eldar_, Oct 09 2022