cp's OEIS Frontend

A385048 The sum of the unitary divisors of n that are cubefull numbers (A036966).

%I A385048 #9 Jun 16 2025 16:53:08
%S A385048 1,1,1,1,1,1,1,9,1,1,1,1,1,1,1,17,1,1,1,1,1,1,1,9,1,1,28,1,1,1,1,33,1,
%T A385048 1,1,1,1,1,1,9,1,1,1,1,1,1,1,17,1,1,1,1,1,28,1,9,1,1,1,1,1,1,1,65,1,1,
%U A385048 1,1,1,1,1,9,1,1,1,1,1,1,1,17,82,1,1,1,1
%N A385048 The sum of the unitary divisors of n that are cubefull numbers (A036966).
%C A385048 The number of these divisors is A368248(n), and the largest of them is A360540(n).
%H A385048 Amiram Eldar, <a href="/A385048/b385048.txt">Table of n, a(n) for n = 1..10000</a>
%F A385048 Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = p^e + 1 if e >= 3.
%F A385048 a(n) = A034448(n) / A371242(n).
%F A385048 a(n) <= A034448(n), with equality if and only if n is cubefull (A036966).
%F A385048 a(n) <= A385005(n), with equality if and only if n is biquadratefree (A046100).
%F A385048 Dirichlet g.f.: zeta(s)*zeta(s-1)*Product_{p prime} (1 - 1/p^(s-1) + 1/p^(3*s-3) - 1/p^(4*s-3)).
%t A385048 f[p_, e_] := If[e <= 2, 1, p^e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A385048 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] < 3, 1, f[i, 1]^f[i, 2] + 1));}
%Y A385048 The unitary analog of A385005.
%Y A385048 Cf. A034448, A036966, A046100, A360540, A368248.
%Y A385048 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), this sequence (cubefull), A385049 (biquadratefree).
%K A385048 nonn,easy,mult
%O A385048 1,8
%A A385048 _Amiram Eldar_, Jun 16 2025