login
A366764
The sum of divisors of n that have no exponent 2 in their prime factorization.
3
1, 3, 4, 3, 6, 12, 8, 11, 4, 18, 12, 12, 14, 24, 24, 27, 18, 12, 20, 18, 32, 36, 24, 44, 6, 42, 31, 24, 30, 72, 32, 59, 48, 54, 48, 12, 38, 60, 56, 66, 42, 96, 44, 36, 24, 72, 48, 108, 8, 18, 72, 42, 54, 93, 72, 88, 80, 90, 60, 72, 62, 96, 32, 123, 84, 144, 68
OFFSET
1,2
COMMENTS
The sum of terms of A337050 that divide n.
The number of these divisors is A366763(n), and the largest of them is A366765(n).
LINKS
FORMULA
Multiplicative with a(p) = p + 1, and a(p^e) = (p^(e+1) - 1)/(p - 1) - p^2 for e >= 2.
a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).
a(n) >= A034448(n), with equality if and only if n is cubefree (A004709).
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^3-p)) = 1.231291... (A065487).
MATHEMATICA
f[p_, e_] := (p^(e + 1) - 1)/(p - 1) - p^2; f[p_, 1] := p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, f[i, 1] + 1, (f[i, 1]^(f[i, 2] + 1) - 1)/(f[i, 1] - 1) - f[i, 1]^2)); }
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Oct 21 2023
STATUS
approved