A380396 - OEIS

%I #10 Jan 25 2025 08:29:33

%S 1,1,1,1,1,1,1,9,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,9,1,1,28,1,1,1,1,1,1,1,

%T 1,1,1,1,1,9,1,1,1,1,1,1,1,1,1,1,1,1,1,28,1,9,1,1,1,1,1,1,1,65,1,1,1,

%U 1,1,1,1,9,1,1,1,1,1,1,1,1,1,1,1,1,1,1

%N a(n) is the sum of the unitary divisors of n that are cubes.

%C The number of unitary divisors of n that are cubes is A380395(n).

%H Amiram Eldar, <a href="/A380396/b380396.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * [d is cube], where [] is the Iverson bracket.

%F a(n) >= 1, with equality if and only if n is not in A366761.

%F a(n) <= A113061(n), with equality if and only if n is biquadratefree (A046100).

%F Multiplicative with a(p^e) = p^e + 1 if e is divisible by 3, and 1 otherwise.

%F Sum_{k=1..n} a(k) ~ c * n^(4/3) / 4, where c = zeta(4/3)/zeta(7/3) = 2.54455250463133711749... .

%F Dirichlet g.f.: zeta(s) * zeta(3*s-3) / zeta(4*s-3).

%F In general, the average order of the sum of the unitary divisors that are m-powers is c * n^(1+1/m) / (m+1), where c = zeta(1+1/m)/zeta(2+1/m), and its Dirichlet g.f. is zeta(s) * zeta(m*s-m) / zeta((m+1)*s-m), both for m >= 2.

%e a(8) = 9 since 8 has 2 unitary divisors that are cubes, 1 = 1^3 and 8 = 2^3, and 1 + 8 = 9.

%e a(216) = 252 since 216 has 4 unitary divisors that are cubes, 1 = 1^3, 8 = 2^3, 27 = 3^3 and 216 = 6^3, and 1 + 8 + 27 + 216 = 252.

%t f[p_, e_] := If[Divisible[e, 3], p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (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 Cf. A000578, A046100, A077610, A113061, A358347, A366761, A371334, A380395.

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Jan 23 2025