OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^e) = (p^(e+1) - 1)/(p - 1) - 1, if e is odd, and (p^(e+1) - 1)/(p - 1) - p^(e/2) - 1 if e is even.
a(n) >= n, with equality if and only if n is cubefree (A004709).
a(n) >= A362852(n), with equality if and only if n = 1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/2) * Product_{p prime} (p/(p+1))*(1+1/p-1/p^3+2/p^5) = 0.557782322450569540209... .
Dirichlet g.f.: zeta(s-1) * zeta(s) * zeta(2*s-1) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s-1) + 1/p^(3*s-2) + 2/p^(3*s-1) - 2/p^(4*s-2)). - Amiram Eldar, Oct 01 2023
EXAMPLE
a(8) = 14 since 8 has 3 divisors that are both bi-unitary and coreful, 2, 4 and 8, and 2 + 4 + 8 = 14.
MATHEMATICA
f[p_, e_] := (p^(e+1) - 1)/(p - 1) - 1 - If[OddQ[e], 0, p^(e/2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - 1 - if(f[i, 2]%2, 0, f[i, 1]^(f[i, 2]/2))); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, May 28 2023
STATUS
approved