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A331737 Multiplicative with a(p^e) = p^A000265(e), where A000265(x) gives the odd part of x.

%I A331737 #13 Nov 17 2022 05:14:35
%S A331737 1,2,3,2,5,6,7,8,3,10,11,6,13,14,15,2,17,6,19,10,21,22,23,24,5,26,27,
%T A331737 14,29,30,31,32,33,34,35,6,37,38,39,40,41,42,43,22,15,46,47,6,7,10,51,
%U A331737 26,53,54,55,56,57,58,59,30,61,62,21,8,65,66,67,34,69,70,71,24,73,74,15,38,77,78,79,10,3,82,83,42,85,86,87,88,89,30
%N A331737 Multiplicative with a(p^e) = p^A000265(e), where A000265(x) gives the odd part of x.
%C A331737 a(n) is the largest exponential divisor of n (cf. A322791) that is an exponentially odd number (A268335). - _Amiram Eldar_, Nov 17 2022
%H A331737 Antti Karttunen, <a href="/A331737/b331737.txt">Table of n, a(n) for n = 1..16384</a>
%H A331737 Brahim Mittou, <a href="https://www.montis.pmf.ac.me/allissues/53/Mathematica-Montisnigri-53-1.pdf">New properties of an arithmetic function</a>, Mathematica Montisnigri, Vol LIII (2022).
%F A331737 a(n) = n / A331738(n).
%F A331737 Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1-1/p) * Sum_{k>=1} p^(2^k - 1)/(p^(2^(k+1)-2) - 1)) = 0.3953728204... . - _Amiram Eldar_, Nov 17 2022
%t A331737 f[p_, e_] := p^(e/2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Nov 17 2022 *)
%o A331737 (PARI)
%o A331737 A000265(n) = (n>>valuation(n,2));
%o A331737 A331737(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^A000265(f[k, 2])); };
%Y A331737 Cf. A000265, A268335, A322791, A331738.
%K A331737 nonn,mult
%O A331737 1,2
%A A331737 _Antti Karttunen_, Feb 02 2020