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A353897 a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).

%I A353897 #14 Aug 24 2023 16:22:39
%S A353897 1,2,3,4,5,6,7,4,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,12,25,26,
%T A353897 9,28,29,30,31,16,33,34,35,36,37,38,39,20,41,42,43,44,45,46,47,48,49,
%U A353897 50,51,52,53,18,55,28,57,58,59,60,61,62,63,16,65,66,67,68
%N A353897 a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).
%H A353897 Amiram Eldar, <a href="/A353897/b353897.txt">Table of n, a(n) for n = 1..10000</a>
%F A353897 Multiplicative with a(p^e) = p^(2^floor(log_2(e))).
%F A353897 a(n) = n if and only if n is in A138302.
%F A353897 Sum_{k=1..n} a(k) ~ c*n^2, where c = 0.4616988732... = (1/2) * Product_{p prime} (1 + Sum_{k>=1} (p^f(k) - p^(f(k-1)+1))/p^(2*k)), f(k) = 2^floor(log_2(k)) and f(0) = 0.
%e A353897 a(27) = 9 since 9 = 3^2 is the largest divisor of 27 with an exponent in its prime factorization, 2, that is a power of 2.
%t A353897 f[p_, e_] := p^(2^Floor[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%Y A353897 Cf. A138302, A353898.
%Y A353897 Similar sequences: A000265, A007947, A008834, A055071, A350390.
%K A353897 nonn,mult
%O A353897 1,2
%A A353897 _Amiram Eldar_, May 10 2022