A383763 The sum of unitary divisors of n that are exponentially squarefree numbers.

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 33, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 65, 84, 144, 68

Offset: 1

Author

Amiram Eldar, May 09 2025

Keywords

• nonn
• easy
• mult

Comments

The number of these divisors is A383762(n) and the largest of them is A383764(n).

Links

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

Crossrefs

Cf. A005117, A034448, A077610, A209061, A365682, A383762, A383764.

Programs

Mathematica

f[p_, e_] := If[SquareFreeQ[e], p^e + 1, 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(issquarefree(f[i,2]), f[i,1]^f[i,2]+1, 1));}

Formula

Multiplicative with a(p^e) = p^e + 1 if e is squarefree (A005117), and 1 otherwise.

a(n) <= A034448(n), with equality if and only if n is an exponentially squarefree number (A209061).

a(n) <= A365682(n), with equality if and only if n is a squarefree number.