A366764 The sum of divisors of n that have no exponent 2 in their prime factorization.

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

Author

Amiram Eldar, Oct 21 2023

Keywords

• nonn
• easy
• mult

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A004709, A005117, A034448, A065487, A337050, A366763, A366765.

Programs

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]

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));}

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).