A380396 a(n) is the sum of the unitary divisors of n that are cubes.
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, 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, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 1
Examples
a(8) = 9 since 8 has 2 unitary divisors that are cubes, 1 = 1^3 and 8 = 2^3, and 1 + 8 = 9. 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.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
f[p_, e_] := If[Divisible[e, 3], p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
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PARI
a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%3, 1, f[i, 1]^f[i, 2] + 1));}
Formula
a(n) = Sum_{d|n, gcd(d, n/d) = 1} d * [d is cube], where [] is the Iverson bracket.
a(n) >= 1, with equality if and only if n is not in A366761.
Multiplicative with a(p^e) = p^e + 1 if e is divisible by 3, and 1 otherwise.
Sum_{k=1..n} a(k) ~ c * n^(4/3) / 4, where c = zeta(4/3)/zeta(7/3) = 2.54455250463133711749... .
Dirichlet g.f.: zeta(s) * zeta(3*s-3) / zeta(4*s-3).
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.
Comments