A383762 -id:A383762 - cp's OEIS frontend

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

Amiram Eldar, May 09 2025

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

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

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

f[p_, e_] := If[SquareFreeQ[e], p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(issquarefree(f[i,2]), f[i,1]^f[i,2]+1, 1));}

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.

A383764 The largest unitary divisor of n that is an exponentially squarefree number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68

Offset: 1

Amiram Eldar, May 09 2025

First differs from A053165 at n = 32.

The number of these divisors is A383762(n) and their sum is A383763(n).

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

Cf. A005117, A053165, A077610, A209061, A365683, A383762, A383763.

f[p_, e_] := If[SquareFreeQ[e], p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(issquarefree(f[i,2]), f[i,1]^f[i,2], 1));}

Multiplicative with a(p^e) = p^e if e is squarefree (A005117), and 1 otherwise.
a(n) <= A365683(n), with equality if and only if n is an exponentially squarefree number (A209061).
a(n) <= n, with equality if and only if n is an exponentially squarefree number.

cp's OEIS Frontend

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

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