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A385049 The sum of the unitary divisors of n that are biquadratefree numbers (A046100).

%I A385049 #9 Jun 16 2025 16:52:53
%S A385049 1,3,4,5,6,12,8,9,10,18,12,20,14,24,24,1,18,30,20,30,32,36,24,36,26,
%T A385049 42,28,40,30,72,32,1,48,54,48,50,38,60,56,54,42,96,44,60,60,72,48,4,
%U A385049 50,78,72,70,54,84,72,72,80,90,60,120,62,96,80,1,84,144,68,90
%N A385049 The sum of the unitary divisors of n that are biquadratefree numbers (A046100).
%C A385049 First differs from A383763 at n = 32.	
%C A385049 The number of these divisors is A365499(n), and the largest of them is A385007(n).
%D A385049 D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.
%H A385049 Amiram Eldar, <a href="/A385049/b385049.txt">Table of n, a(n) for n = 1..10000</a>
%H A385049 Francesco Pappalardi, <a href="http://www.mat.uniroma3.it/users/pappa/papers/allahabad2003.pdf">A survey on k-freeness</a>, Number Theory, Ramanujan Math. Soc. Lect. Notes Ser., Vol. 1 (2003), pp. 71-88.
%F A385049 Multiplicative with a(p^e) = p^e + 1 for e <= 3, and a(p^e) = 1 for e >= 4.
%F A385049 a(n) = 1 if and only if n is 4-full (A036967).
%F A385049 a(n) <= A034448(n), with equality if and only if n is biquadratefree.
%F A385049 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2) - 1/p^(4*s-3)).
%F A385049 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2 + p) - 1/p^4) = 1.27769267395905900191... .
%t A385049 f[p_, e_] := If[e < 4, p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385049 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 4, f[i, 1]^f[i, 2] + 1, 1)); }
%Y A385049 The unitary analog of A385006.
%Y A385049 Cf. A036967, A046100, A365499, A385007.
%Y A385049 The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), this sequence (biquadratefree).
%K A385049 nonn,easy,mult
%O A385049 1,2
%A A385049 _Amiram Eldar_, Jun 16 2025