%I
%S 1,2,4,3,3,8,5,5,16,9,6,6,9,32,17,10,7,7,10,7,7,17,7,7,64,33,18,12,11,
%T 11,12,18,11,11,33,11,11,128,65,34,20,19,19,13,13,20,13,13,34,19,19,
%U 65,19,13,13,19,256,129,66,36,35
%N Sum of parts of the form 10...0 with nonnegative number of zeros in binary representation of csquarefree numbers (A233564) as the corresponding powers of 2.
%C Subsequence of A162439.
%C The number of times of appearances of number k in the sequence is the number of compositions of k into distinct powers of 2, i.e., it is A000120(k)!
%F Let, for k_1>k_2>...>k_r, A233564(n) = 2^k_1 + 2^k_2 +...+ 2^k_r. Then a(n) = 2^(k_1k_21) + 2^(k_2k_31) + 2^(k_(r1)k_r1) + 2^k_r.
%e Let n=17, A233564(17)=37. In binary a concatenation of parts of the form 10...0 which gives 37 is (100)(10)(1). Thus a(17)= 4+2+1 = 7.
%t bitPatt[n_]:=bitPatt[n]=Split[IntegerDigits[n,2],#2==0&]; Map[Plus@@(Map[FromDigits[#,2]&,bitPatt[#]])&,Select[Range[300],#==DeleteDuplicates[#]&[bitPatt[#]]&]] (* _Peter J. C. Moses_, Jan 15 2014 *)
%Y Cf. A233564, A162439, A000120.
%K nonn,base
%O 1,2
%A _Vladimir Shevelev_, Jan 12 2014
