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A365170 The sum of divisors d of n such that gcd(d, n/d) is squarefree.

%I A365170 #10 May 24 2024 04:36:42
%S A365170 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,27,18,39,20,42,32,36,24,60,31,
%T A365170 42,40,56,30,72,32,51,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,
%U A365170 108,57,93,72,98,54,120,72,120,80,90,60,168,62,96,104,99,84,144
%N A365170 The sum of divisors d of n such that gcd(d, n/d) is squarefree.
%C A365170 The number of these divisors is A252505(n).
%H A365170 Amiram Eldar, <a href="/A365170/b365170.txt">Table of n, a(n) for n = 1..10000</a>
%F A365170 Multiplicative with a(p) = 1 + p, a(p^2) = 1 + p + p^2, and a(p^e) = (1 + p)*(1 + p^(e - 1)) if e >= 3.
%F A365170 a(n) >= A034448(n), with equality if and only if n is squarefree number (A005117).
%F A365170 a(n) <= A000203(n), with equality if and only if n is biquadratefree number (A046100).
%F A365170 Sum_{k=1..n} a(k) ~ c * n^2, where c = 315/(4*Pi^4) = A157292 / 2 = 0.808446... .
%t A365170 f[p_, e_] := Switch[e, 1, 1 + p, 2, 1 + p + p^2, _, (1 + p)*(1 + p^(e - 1))]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365170 (PARI) a(n) = {my(f = factor(n), p , e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; if(e == 1, 1 + p, if(e == 2, 1 + p + p^2, (1 + p)*(1 + p^(e - 1)))));}
%Y A365170 Cf. A000203, A000290, A005117, A034448, A046100, A157292, A252505.
%K A365170 nonn,easy,mult
%O A365170 1,2
%A A365170 _Amiram Eldar_, Aug 25 2023