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

%I A357689 #34 Jul 13 2025 01:17:21
%S A357689 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 A357689 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 A357689 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 A357689 a(n) = n/A204455(n), where A204455(n) is the product of odd noncomposite divisors of n.
%H A357689 Michael De Vlieger, <a href="/A357689/b357689.txt">Table of n, a(n) for n = 1..10000</a>
%F A357689 a(n)*A204455(n) = n.
%F A357689 a(n) = EvenPart(n)*A003557(OddPart(n)). - _Peter Munn_, Oct 09 2022
%F A357689 Multiplicative with a(p^e) = p^(e-1) if p > 2 and a(2^e) = 2^e. - _Amiram Eldar_, Oct 10 2022
%F A357689 a(n) = A003557(2*n). - _Ridouane Oudra_, Jul 12 2025
%e A357689 n = A204455(n)*a(n): 1 = 1*1, 2 = 1*2, 3 = 3*1, 4 = 1*4, 5 = 5*1, 6 = 3*2.
%t A357689 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 A357689 (Magma) [(2*n)/&*PrimeDivisors(2*n): n in [1..100]];
%o A357689 (Magma) [n/&*[d: d in Divisors(n) | d*#[m: m in [0..n-1] | -m^d mod d eq m] eq
%o A357689   #[m: m in [0..n-1] | m^d mod d eq m]]: n in [1..100]];
%Y A357689 Cf. A000265, A003557, A006519, A204455.
%Y A357689 Equals A324873 up to a(32) = 32.
%K A357689 nonn,mult,easy
%O A357689 1,2
%A A357689 _Juri-Stepan Gerasimov_, Oct 09 2022