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

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

Author

Amiram Eldar, May 09 2025

Keywords

• nonn
• easy
• mult

Comments

First differs from A053165 at n = 32.

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

Links

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

Crossrefs

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

Programs

Mathematica

f[p_, e_] := If[SquareFreeQ[e], p^e, 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));}

Formula

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.