A368167 The largest unitary divisor of n that is a cubefull exponentially odd number (A335988).

1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 27, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 27, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Dec 14 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A056191 and A366126 at n = 32, and from A367513 at n = 64.

Also, the largest exponentially odd unitary divisor of the powerful part on n.

Also, the powerful part of the largest exponentially odd unitary divisor of n.

Links

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

Crossrefs

Cf. A057521, A077610, A335275, A335988, A350389, A368168, A368169.

Cf. A056191, A366126, A367513.

Programs

Mathematica

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

PARI

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

Formula

Multiplicative with a(p^e) = p^e if e is odd that is larger than 1, and 1 otherwise.

a(n) = A350389(A057521(n)).

a(n) = A057521(A350389(n)).

a(n) >= 1, with equality if and only if n is in A335275.

a(n) <= n, with equality if and only if n is in A335988.