sum of unit fractions

[David C Terr]

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.

Dave






hv at crypt.org
12/08/2004 03:44 PM

 
        To:     seqfan at ext.jussieu.fr
        cc: 
        Subject:        Re: sum of unit fractions

hv at crypt.org wrote:
:Define a(n) as the least integer k such that there is a sum of distinct 
unit
:fractions equal to _n_ of which the greatest denominator is k.
:
:Alternatively:
:  a(n) = k implies that there exists a set of positive integers S such 
that
:  sum{1/s_i} = n, and max(S) = k, and no set S' exists with the same sum
:  but a smaller maximal element.
:
:eg a(1) = 1 (S = { 1 })
:   a(2) = 6 (S = { 1 2 3 6 })
:   a(3) = 24 (S = { 1 2 3 4 5 6 8 9 10 15 18 20 24 })
[...]
:Working by hand, I believe I've established a(4) = 65, with the set {
:  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 18 20 22 24
:  26 27 28 30 33 35 36 40 42 45 48 52 54 56 60 63 65
:} from which the only optional discards are { 21 39 44 55 }.

My best attempt for a(5) is 184, using this set: {
  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
  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
  68 69 70 72 75 76 77 78 80 81 84 85 87 88 90 91 92 93 95 96 99 102 104 
105
  108 110 112 114 115 116 117 126 130 133 136 138 140 143 144 145 150 152 
153
  154 155 156 161 162 165 168 170 171 174 175 176 180 184
}

.. and for a(6) is 475: {
  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
  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
  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
  84 85 86 87 88 90 91 92 93 94 95 96 98 99 100 102 104 105 106 108 110 
111
  112 114 115 116 117 119 120 121 122 123 124 126 128 129 130 132 133 134 
135
  136 138 140 141 143 144 145 147 148 150 152 153 154 155 156 159 160 161 
162
  164 165 168 170 171 174 175 176 177 180 182 183 184 185 186 187 188 189 
190
  192 195 196 198 200 201 203 205 207 208 209 210 212 215 216 217 220 221 
222
  224 225 228 230 231 232 234 238 240 242 245 246 247 248 250 252 253 255 
258
  259 260 261 264 266 268 270 272 273 275 276 280 282 285 286 287 288 290 
294
  295 297 299 300 301 304 305 306 308 310 312 315 319 320 322 323 324 325 
328
  329 330 336 340 341 342 344 345 348 350 351 352 354 357 363 364 370 372 
374
  375 376 377 378 384 387 390 391 396 402 405 406 407 408 413 414 416 418 
420
  424 425 429 430 432 434 435 437 442 444 448 450 451 455 456 460 462 465 
468
  469 475
}

My figures for a(4) .. a(6) should be viewed as best known upper bounds
unless and until someone can confirm them.

Hugo


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