A367418 The exponentially odd numbers (A268335) divided by their squarefree kernels (A007947).

1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 9, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 9, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1

Offset: 1

Author

Amiram Eldar, Nov 17 2023

Keywords

• nonn
• easy

Comments

Analogous to A102631, with the exponentially odd numbers instead of the square numbers (A000290).

All the terms are square numbers.

Links

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

Crossrefs

Cf. A000290, A008833, A102631, A268335, A367406, A367417, A367419.

Programs

Mathematica

s[n_] := n / Times @@ FactorInteger[n][[;; , 1]]; s /@ Select[Range[200], AllTrue[FactorInteger[#][[;; , 2]], OddQ] &]

PARI

b(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, f[i, 1]^(f[i, 2]-1), 0)); }
lista(kmax) = {my(b1); for(k = 1, kmax, b1 = b(k); if(b1 > 0, print1(b1, ", "))); }

Formula

a(n) = A003557(A268335(n)).

a(n) = A268335(n)/A367417(n).

a(n) = A367419(n)^2.

a(n) = A268335(n)^2/A367406(n).

a(n) = A008833(A268335(n)). - Amiram Eldar, Nov 30 2023