%I #9 Jan 08 2019 16:23:28
%S 1,1,2,2,4,4,6,9,12,13,18,23,29,37,49,55,71,84,104,126,153,185,221,
%T 261,317,375,446,523,623,721,854,994,1168,1357,1579,1833,2126,2455,
%U 2843,3270,3766,4320,4980,5687,6521,7444,8498,9684,11039,12540,14262
%N Number of integer partitions of n where no part is 2^k times any other part, for any k > 0.
%e The a(1) = 1 through a(8) = 12 integer partitions:
%e (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (111) (22) (32) (33) (43) (44)
%e (31) (311) (51) (52) (53)
%e (1111) (11111) (222) (61) (62)
%e (3111) (322) (71)
%e (111111) (331) (332)
%e (511) (611)
%e (31111) (2222)
%e (1111111) (3311)
%e (5111)
%e (311111)
%e (11111111)
%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t Table[Length[Select[IntegerPartitions[n],stableQ[#,IntegerQ[Log[2,#1/#2]]&]&]],{n,30}]
%Y Cf. A018819, A101417, A120641, A276431, A305148, A323092, A323094.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jan 04 2019