Comment from R. J. Mathar, Jul 13 2008:
Examples of a(n), k, and the two binary representations of a(n) and a(n)*k:
(Not necessarily complete!)

11 3 [1, 1, 0, 1] [1, 0, 0, 0, 0, 1]
13 5 [1, 0, 1, 1] [1, 0, 0, 0, 0, 0, 1]
19 27 [1, 1, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
22 3 [0, 1, 1, 0, 1] [0, 1, 0, 0, 0, 0, 1]
23 3 [1, 1, 1, 0, 1] [1, 0, 1, 0, 0, 0, 1]
25 41 [1, 0, 0, 1, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
26 5 [0, 1, 0, 1, 1] [0, 1, 0, 0, 0, 0, 0, 1]
27 3 [1, 1, 0, 1, 1] [1, 0, 0, 0, 1, 0, 1]
29 5 [1, 0, 1, 1, 1] [1, 0, 0, 0, 1, 0, 0, 1]
37 7085 [1, 0, 1, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
38 27 [0, 1, 1, 0, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
39 7 [1, 1, 1, 0, 0, 1] [1, 0, 0, 0, 1, 0, 0, 0, 1]
41 25 [1, 0, 0, 1, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
43 3 [1, 1, 0, 1, 0, 1] [1, 0, 0, 0, 0, 0, 0, 1]
44 3 [0, 0, 1, 1, 0, 1] [0, 0, 1, 0, 0, 0, 0, 1]
46 3 [0, 1, 1, 1, 0, 1] [0, 1, 0, 1, 0, 0, 0, 1]
47 3 [1, 1, 1, 1, 0, 1] [1, 0, 1, 1, 0, 0, 0, 1]
50 41 [0, 1, 0, 0, 1, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
52 5 [0, 0, 1, 0, 1, 1] [0, 0, 1, 0, 0, 0, 0, 0, 1]
53 5 [1, 0, 1, 0, 1, 1] [1, 0, 0, 1, 0, 0, 0, 0, 1]
54 3 [0, 1, 1, 0, 1, 1] [0, 1, 0, 0, 0, 1, 0, 1]
55 3 [1, 1, 1, 0, 1, 1] [1, 0, 1, 0, 0, 1, 0, 1]
57 9 [1, 0, 0, 1, 1, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
58 5 [0, 1, 0, 1, 1, 1] [0, 1, 0, 0, 0, 1, 0, 0, 1]
59 3 [1, 1, 0, 1, 1, 1] [1, 0, 0, 0, 1, 1, 0, 1]
61 5 [1, 0, 1, 1, 1, 1] [1, 0, 0, 0, 1, 1, 0, 0, 1]
71 119 [1, 1, 1, 0, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
74 7085 [0, 1, 0, 1, 0, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
76 27 [0, 0, 1, 1, 0, 0, 1] [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
77 5 [1, 0, 1, 1, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 1, 1]
78 7 [0, 1, 1, 1, 0, 0, 1] [0, 1, 0, 0, 0, 1, 0, 0, 0, 1]
79 7 [1, 1, 1, 1, 0, 0, 1] [1, 0, 0, 1, 0, 1, 0, 0, 0, 1]
82 25 [0, 1, 0, 0, 1, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
83 395 [1, 1, 0, 0, 1, 0, 1] [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
86 3 [0, 1, 1, 0, 1, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 1]
87 3 [1, 1, 1, 0, 1, 0, 1] [1, 0, 1, 0, 0, 0, 0, 0, 1]
88 3 [0, 0, 0, 1, 1, 0, 1] [0, 0, 0, 1, 0, 0, 0, 0, 1]
91 3 [1, 1, 0, 1, 1, 0, 1] [1, 0, 0, 0, 1, 0, 0, 0, 1]
92 3 [0, 0, 1, 1, 1, 0, 1] [0, 0, 1, 0, 1, 0, 0, 0, 1]
94 3 [0, 1, 1, 1, 1, 0, 1] [0, 1, 0, 1, 1, 0, 0, 0, 1]
95 3 [1, 1, 1, 1, 1, 0, 1] [1, 0, 1, 1, 1, 0, 0, 0, 1]
99 11 [1, 1, 0, 0, 0, 1, 1] [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1]
100 41 [0, 0, 1, 0, 0, 1, 1] [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
101 365 [1, 0, 1, 0, 0, 1, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1]
103 5 [1, 1, 1, 0, 0, 1, 1] [1, 1, 0, 0, 0, 0, 0, 0, 0, 1]
104 5 [0, 0, 0, 1, 0, 1, 1] [0, 0, 0, 1, 0, 0, 0, 0, 0, 1]
106 5 [0, 1, 0, 1, 0, 1, 1] [0, 1, 0, 0, 1, 0, 0, 0, 0, 1]
107 3 [1, 1, 0, 1, 0, 1, 1] [1, 0, 0, 0, 0, 0, 1, 0, 1]
108 3 [0, 0, 1, 1, 0, 1, 1] [0, 0, 1, 0, 0, 0, 1, 0, 1]
109 5 [1, 0, 1, 1, 0, 1, 1] [1, 0, 0, 0, 0, 1, 0, 0, 0, 1]
110 3 [0, 1, 1, 1, 0, 1, 1] [0, 1, 0, 1, 0, 0, 1, 0, 1]
111 3 [1, 1, 1, 1, 0, 1, 1] [1, 0, 1, 1, 0, 0, 1, 0, 1]
113 145 [1, 0, 0, 0, 1, 1, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
114 9 [0, 1, 0, 0, 1, 1, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1]
115 9 [1, 1, 0, 0, 1, 1, 1] [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1]
116 5 [0, 0, 1, 0, 1, 1, 1] [0, 0, 1, 0, 0, 0, 1, 0, 0, 1]
117 5 [1, 0, 1, 0, 1, 1, 1] [1, 0, 0, 1, 0, 0, 1, 0, 0, 1]
118 3 [0, 1, 1, 0, 1, 1, 1] [0, 1, 0, 0, 0, 1, 1, 0, 1]
119 3 [1, 1, 1, 0, 1, 1, 1] [1, 0, 1, 0, 0, 1, 1, 0, 1]
121 9 [1, 0, 0, 1, 1, 1, 1] [1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1]
122 5 [0, 1, 0, 1, 1, 1, 1] [0, 1, 0, 0, 0, 1, 1, 0, 0, 1]
123 3 [1, 1, 0, 1, 1, 1, 1] [1, 0, 0, 0, 1, 1, 1, 0, 1]
125 5 [1, 0, 1, 1, 1, 1, 1] [1, 0, 0, 0, 1, 1, 1, 0, 0, 1]
139 59 [1, 1, 0, 1, 0, 0, 0, 1] [1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
141 581 [1, 0, 1, 1, 0, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1]
142 119 [0, 1, 1, 1, 0, 0, 0, 1] [0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]
143 15 [1, 1, 1, 1, 0, 0, 0, 1] [1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1]
145 113 [1, 0, 0, 0, 1, 0, 0, 1] [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
147 7 [1, 1, 0, 0, 1, 0, 0, 1] [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1]
148 7085 [0, 0, 1, 0, 1, 0, 0, 1] [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]
149 55 [1, 0, 1, 0, 1, 0, 0, 1] [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]

The list above is constructed with the naive:
A000120 := proc(n)
        add(i,i=convert(n,base,2)) ;
end:
isA005360 :=  proc(n)
        local k,n120 ;
        n120 := A000120(n) ;
        for k from 2 to 30000 do
                if A000120(n*k) < n120 then
                        RETURN(k) ;
                fi ;
        od:
        RETURN(-1) ;
end:
for n from 1 to 150 do
        a := isA005360(n) ;
        if a > 0 then
                printf("%d %d %a %a\n",n,a,convert(n,base,2),convert(n*a,base,2)) ;
        fi ;
od: