A365170 The sum of divisors d of n such that gcd(d, n/d) is squarefree.

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 27, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 51, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 108, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 99, 84, 144

Offset: 1

Author

Amiram Eldar, Aug 25 2023

Keywords

• nonn
• easy
• mult

Comments

The number of these divisors is A252505(n).

Links

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

Crossrefs

Cf. A000203, A000290, A005117, A034448, A046100, A157292, A252505.

Programs

Mathematica

f[p_, e_] := Switch[e, 1, 1 + p, 2, 1 + p + p^2, , (1 + p)*(1 + p^(e - 1))]; 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]; if(e == 1, 1 + p, if(e == 2, 1 + p + p^2, (1 + p)*(1 + p^(e - 1)))));}

Formula

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

a(n) >= A034448(n), with equality if and only if n is squarefree number (A005117).

a(n) <= A000203(n), with equality if and only if n is biquadratefree number (A046100).

Sum_{k=1..n} a(k) ~ c * n^2, where c = 315/(4*Pi^4) = A157292 / 2 = 0.808446... .