A385134 The sum of divisors d of n such that n/d is a biquadratefree number (A046100).

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 30, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 60, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 120, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 120, 84, 144

Offset: 1

Author

Amiram Eldar, Jun 19 2025

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A000203, A008683, A046100, A307430, A385006.

The sum of divisors d of n such that n/d is: A001615 (squarefree), A002131 (odd), A069208 (powerful), A076752 (square), A129527 (power of 2), A254981 (cubefree), A244963 (nonsquarefree), A327626 (cube), this sequence (biquadratefree), A385135 (exponentially odd), A385136 (cubefull), A385137 (3-smooth), A385138 (5-rough), A385139 (exponentially 2^n).

Programs

Mathematica

f[p_, e_] := p^(e-3)*(1 + p + p^2 + p^3); f[p_, 1] := 1 + p; f[p_, 2] := 1 + p + p^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^max(e-3,0) * (p^min(e+1,4)-1)/(p-1));}

Formula

a(n) = Sum_{d | n} d * A307430(n/d) = n * Sum_{d | n} A307430(d) / d.

a(n) = Sum_{d^3 | n} mu(d) * A000203(n/d^3), where mu is the Moebius function (A008683).

Multiplicative with a(p) = 1 + p, a(p^2) = 1 + p + p^2, and a(p^e) = p^(e-3) * (1 + p + p^2 + p^3), for e >= 3.

In general, the sum of divisors d of n such that n/d is k-free (not divisible by a k-th power larger than 1) is multiplicative with a(p^e) = p^max(e-k+1,0) * (p^min(e+1,k)-1)/(p-1).

Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(4*s).

In general, the sum of divisors d of n such that n/d is k-free has Dirichlet g.f.: zeta(s) * zeta(s-1) / zeta(k*s).

Sum_{i=1..n} a(i) ~ (1575 / (2*Pi^6)) * n^2.