A357689 a(n) = n/A204455(n), where A204455(n) is the product of odd noncomposite divisors of n.

1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, 1, 16, 1, 6, 1, 4, 1, 2, 1, 8, 5, 2, 9, 4, 1, 2, 1, 32, 1, 2, 1, 12, 1, 2, 1, 8, 1, 2, 1, 4, 3, 2, 1, 16, 7, 10, 1, 4, 1, 18, 1, 8, 1, 2, 1, 4, 1, 2, 3, 64, 1, 2, 1, 4, 1, 2, 1, 24, 1, 2, 5, 4, 1, 2, 1, 16, 27, 2, 1, 4, 1, 2, 1, 8, 1, 6, 1, 4, 1, 2, 1, 32, 1, 14, 3, 20

Offset: 1

Author

Juri-Stepan Gerasimov, Oct 09 2022

Keywords

• nonn
• mult
• easy

Examples

			n = A204455(n)*a(n): 1 = 1*1, 2 = 1*2, 3 = 3*1, 4 = 1*4, 5 = 5*1, 6 = 3*2.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000265, A003557, A006519, A204455.

Equals A324873 up to a(32) = 32.

Programs

Magma

[(2*n)/&*PrimeDivisors(2*n): n in [1..100]];

Magma

[n/&*[d: d in Divisors(n) | d*#[m: m in [0..n-1] | -m^d mod d eq m] eq
  #[m: m in [0..n-1] | m^d mod d eq m]]: n in [1..100]];

Mathematica

f[p_, e_] := p^(e - If[p == 2, 0, 1]); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 10 2022 *)

Formula

a(n)*A204455(n) = n.

a(n) = EvenPart(n)*A003557(OddPart(n)). - Peter Munn, Oct 09 2022

Multiplicative with a(p^e) = p^(e-1) if p > 2 and a(2^e) = 2^e. - Amiram Eldar, Oct 10 2022

a(n) = A003557(2*n). - Ridouane Oudra, Jul 12 2025