<br><font size=2 face="sans-serif">This looks like quite an interesting problem! One can of course generalize it to arbitrary rational numbers. For instance, a(3/2) = 2 (S = {1,2}). I'd be interested to see how the Farey tree gets mapped by a.</font>
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<br><font size=2 face="sans-serif">Dave</font>
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<td><font size=1 face="sans-serif"><b>hv@crypt.org</b></font>
<p><font size=1 face="sans-serif">12/08/2004 03:44 PM</font>
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<br><font size=1 face="sans-serif"> To: seqfan@ext.jussieu.fr</font>
<br><font size=1 face="sans-serif"> cc: </font>
<br><font size=1 face="sans-serif"> Subject: Re: sum of unit fractions</font></table>
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<br><font size=2 face="Courier New">hv@crypt.org wrote:<br>
:Define a(n) as the least integer k such that there is a sum of distinct unit<br>
:fractions equal to _n_ of which the greatest denominator is k.<br>
:<br>
:Alternatively:<br>
: a(n) = k implies that there exists a set of positive integers S such that<br>
: sum{1/s_i} = n, and max(S) = k, and no set S' exists with the same sum<br>
: but a smaller maximal element.<br>
:<br>
:eg a(1) = 1 (S = { 1 })<br>
: a(2) = 6 (S = { 1 2 3 6 })<br>
: a(3) = 24 (S = { 1 2 3 4 5 6 8 9 10 15 18 20 24 })<br>
[...]<br>
:Working by hand, I believe I've established a(4) = 65, with the set {<br>
: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 22 24<br>
: 26 27 28 30 33 35 36 40 42 45 48 52 54 56 60 63 65<br>
:} from which the only optional discards are { 21 39 44 55 }.<br>
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My best attempt for a(5) is 184, using this set: {<br>
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28<br>
30 32 33 34 35 36 38 39 40 42 44 45 48 50 51 52 54 55 56 58 60 62 63 65 66<br>
68 69 70 72 75 76 77 78 80 81 84 85 87 88 90 91 92 93 95 96 99 102 104 105<br>
108 110 112 114 115 116 117 126 130 133 136 138 140 143 144 145 150 152 153<br>
154 155 156 161 162 165 168 170 171 174 175 176 180 184<br>
}<br>
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.. and for a(6) is 475: {<br>
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28<br>
29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53<br>
54 55 56 57 58 60 61 62 63 64 65 66 67 68 69 70 72 74 75 76 77 78 80 81 82<br>
84 85 86 87 88 90 91 92 93 94 95 96 98 99 100 102 104 105 106 108 110 111<br>
112 114 115 116 117 119 120 121 122 123 124 126 128 129 130 132 133 134 135<br>
136 138 140 141 143 144 145 147 148 150 152 153 154 155 156 159 160 161 162<br>
164 165 168 170 171 174 175 176 177 180 182 183 184 185 186 187 188 189 190<br>
192 195 196 198 200 201 203 205 207 208 209 210 212 215 216 217 220 221 222<br>
224 225 228 230 231 232 234 238 240 242 245 246 247 248 250 252 253 255 258<br>
259 260 261 264 266 268 270 272 273 275 276 280 282 285 286 287 288 290 294<br>
295 297 299 300 301 304 305 306 308 310 312 315 319 320 322 323 324 325 328<br>
329 330 336 340 341 342 344 345 348 350 351 352 354 357 363 364 370 372 374<br>
375 376 377 378 384 387 390 391 396 402 405 406 407 408 413 414 416 418 420<br>
424 425 429 430 432 434 435 437 442 444 448 450 451 455 456 460 462 465 468<br>
469 475<br>
}<br>
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My figures for a(4) .. a(6) should be viewed as best known upper bounds<br>
unless and until someone can confirm them.<br>
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Hugo<br>
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