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a(n) = n/A204455(n), where A204455(n) is the product of odd noncomposite divisors of n.
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%I #30 Oct 18 2022 03:40:14

%S 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,

%T 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,

%U 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

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

%H Michael De Vlieger, <a href="/A357689/b357689.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = EvenPart(n)*A003557(OddPart(n)). - _Peter Munn_, Oct 09 2022

%F Multiplicative with a(p^e) = p^(e-1) if p > 2 and a(2^e) = 2^e. - _Amiram Eldar_, Oct 10 2022

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

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

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

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

%o #[m: m in [0..n-1] | m^d mod d eq m]]: n in [1..100]];

%Y Cf. A000265, A003557, A006519, A204455.

%Y Equals A324873 up to a(32) = 32.

%K nonn,mult,easy

%O 1,2

%A _Juri-Stepan Gerasimov_, Oct 09 2022