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%I #5 May 20 2013 00:49:27
%S 1,2,4,8,16,32,64,80,128,160,208,256,320,416,512,640,832,1024,1280,
%T 1344,1664,2048,2560,2688,3328,4096,5120,5376,6656,8192,10240,10752,
%U 13312,16384,20480,21504,21760,26624,27264,32768,40960,43008,43520,53248,54528
%N Numbers n such that the products n*(sum of the reciprocals of the Collatz (3x+1) sequence beginning at n) are integers.
%C Numbers n such that A225784(n) divides n.
%C The powers of 2 are in the sequence, but there exists a subsequence of non-powers of 2: 80, 160, 208, 320, 416, 640, 832, 1280, 1344,... where the members are of the forms 5*2^p with p>=4, 13*2^p with p>=4, 21*2^p with p>=6, 213*2^p with p>=7, 341*2^p with p>=10,...
%e 208 is in the sequence because 208 *(1/208 + 1/104 + 1/52 + 1/26 + 1/13 + 1/40 + 1/20 + 1/10 + 1/5 + 1/16 + 1/8 + 1/4 +1/2 + 1/1) = 512 is integer.
%e 2688 is in the sequence because A225784(2688) = 896 divides 2688.
%t collatz[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&];Select[Range[50000],IntegerQ[Total[#/collatz[#]]]&]
%Y Cf. A225784, A225761.
%K nonn
%O 1,2
%A _Michel Lagneau_, May 19 2013
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