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: