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Multiplicative with a(p^e) = p^(e-A000265(e)), where A000265(x) gives the odd part of x.
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%I #11 Nov 24 2022 04:20:44

%S 1,1,1,2,1,1,1,1,3,1,1,2,1,1,1,8,1,3,1,2,1,1,1,1,5,1,1,2,1,1,1,1,1,1,

%T 1,6,1,1,1,1,1,1,1,2,3,1,1,8,7,5,1,2,1,1,1,1,1,1,1,2,1,1,3,8,1,1,1,2,

%U 1,1,1,3,1,1,5,2,1,1,1,8,27,1,1,2,1,1,1,1,1,3,1,2,1,1,1,1,1,7,3,10,1,1,1,1,1

%N Multiplicative with a(p^e) = p^(e-A000265(e)), where A000265(x) gives the odd part of x.

%H Antti Karttunen, <a href="/A331738/b331738.txt">Table of n, a(n) for n = 1..20736</a>

%H Antti Karttunen, <a href="/A331738/a331738.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F Multiplicative with a(p^e) = p^A331739(e).

%F a(n) = n / A331737(n).

%t f[p_, e_] := p^(e - e/2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 24 2022 *)

%o (PARI)

%o A000265(n) = (n>>valuation(n,2));

%o A331738(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^(f[k, 2]-A000265(f[k, 2]))); };

%Y Cf. A000265, A331737, A331739.

%K nonn,mult

%O 1,4

%A _Antti Karttunen_, Feb 02 2020