A368540 The smallest unitary divisor d of n such that n/d is a term of A138302.

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, 64, 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, 1, 1, 1, 1, 1, 32, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 27, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 125, 1, 1

Offset: 1

Author

Amiram Eldar, Dec 29 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A368167 at n = 64 and from A367513 at n = 128.

Links

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

Crossrefs

Cf. A048298, A077610, A138302, A367168, A367513, A368167.

Programs

Mathematica

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], 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 << valuation(f[i, 2], 2), 1, f[i, 1]^f[i, 2]));}

Python

from math import prod
from sympy import factorint
def A368540(n): return prod(p**e for p, e in factorint(n).items() if not e or (e&-e)^e) # Chai Wah Wu, Dec 30 2023

Formula

a(n) = n / A367168(n).

Multiplicative with a(p^e) = p^(e-A048298(e)).

a(n) >= 1, with equality if and only if n is in A138302.