%I #14 Aug 24 2023 16:22:39
%S 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 9,28,29,30,31,16,33,34,35,36,37,38,39,20,41,42,43,44,45,46,47,48,49,
%U 50,51,52,53,18,55,28,57,58,59,60,61,62,63,16,65,66,67,68
%N a(n) is the largest divisor of n whose exponents in its prime factorization are all powers of 2 (A138302).
%H Amiram Eldar, <a href="/A353897/b353897.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p^e) = p^(2^floor(log_2(e))).
%F a(n) = n if and only if n is in A138302.
%F 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 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 f[p_, e_] := p^(2^Floor[Log2[e]]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%Y Cf. A138302, A353898.
%Y Similar sequences: A000265, A007947, A008834, A055071, A350390.
%K nonn,mult
%O 1,2
%A _Amiram Eldar_, May 10 2022