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A385134 The sum of divisors d of n such that n/d is a biquadratefree number (A046100).

%I A385134 #12 Jun 20 2025 08:09:32
%S A385134 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,30,18,39,20,42,32,36,24,60,31,
%T A385134 42,40,56,30,72,32,60,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,
%U A385134 120,57,93,72,98,54,120,72,120,80,90,60,168,62,96,104,120,84,144
%N A385134 The sum of divisors d of n such that n/d is a biquadratefree number (A046100).
%H A385134 Amiram Eldar, <a href="/A385134/b385134.txt">Table of n, a(n) for n = 1..10000</a>
%F A385134 a(n) = Sum_{d | n} d * A307430(n/d) = n * Sum_{d | n} A307430(d) / d.
%F A385134 a(n) = Sum_{d^3 | n} mu(d) * A000203(n/d^3), where mu is the Moebius function (A008683).
%F A385134 Multiplicative with a(p) = 1 + p, a(p^2) = 1 + p + p^2, and a(p^e) = p^(e-3) * (1 + p + p^2 + p^3), for e >= 3.
%F A385134 In general, the sum of divisors d of n such that n/d is k-free (not divisible by a k-th power larger than 1) is multiplicative with a(p^e) =  p^max(e-k+1,0) * (p^min(e+1,k)-1)/(p-1).
%F A385134 Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(4*s).
%F A385134 In general, the sum of divisors d of n such that n/d is k-free has Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(k*s).
%F A385134 Sum_{i=1..n} a(i) ~ (1575 / (2*Pi^6)) * n^2.
%t A385134 f[p_, e_] := p^(e-3)*(1 + p + p^2 + p^3); f[p_, 1] := 1 + p; f[p_, 2] := 1 + p + p^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385134 (PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^max(e-3,0) * (p^min(e+1,4)-1)/(p-1));}
%Y A385134 Cf. A000203, A008683, A046100, A307430, A385006.
%Y A385134 The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), this sequence (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).
%K A385134 nonn,easy,mult
%O A385134 1,2
%A A385134 _Amiram Eldar_, Jun 19 2025