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%I #53 Apr 20 2024 23:50:50
%S 1,6,24,65,184,469,1243,3231
%N a(n) = k implies that there exists a set S of positive integers such that Sum_{ s_i in S } 1/s_i = n, max(S) = k and no set S' exists with the same sum and a smaller maximal element.
%C In other words, a(n) = min { max S | S subset of N*, Sum_{x in S} 1/x = n }.
%C 8492 < a(9) <= 8507; 22788 < a(10) <= 22820; 60577 < a(11) <= 60639. - _Hugo van der Sanden_, Jan 20 2015
%C Little is known about the number of different sets S that achieve a(n). Paul Hanna asks if it is always true that a solution set S for n+1 must necessarily contain a solution set for n as a subset. This is true for small n, apparently, but seems to me unlikely to hold in general. - _N. J. A. Sloane_, Dec 31 2005
%C Martin and Shi show that it is not true for n = 5. They give a set that achieves a(6) without 136, while all sets that achieve a(5) include 136. - _Matthieu Pluntz_, Feb 20 2023
%C Does a(n+1) / a(n) converge? - _David A. Corneth_, Apr 08 2018
%C From _Jon E. Schoenfield_, Apr 20 2024: (Start)
%C Does log(a(n)) approach n? Using the known exact values and the bounds from _Hugo van der Sanden_ gives
%C .
%C n log(a(n))
%C == ==========================
%C 1 0
%C 2 1.79175...
%C 3 3.17805...
%C 4 4.17438...
%C 5 5.21493...
%C 6 6.15060...
%C 7 7.12528...
%C 8 8.08054...
%C 9 9.04782... +/- 0.00082...
%C 10 10.03471... +/- 0.00067...
%C 11 11.01219... +/- 0.00050...
%C (End)
%H Greg Martin and Yue Shi, <a href="https://arxiv.org/abs/2107.05076">An algorithm for Egyptian fraction representations with restricted denominators</a>, arXiv:2107.05076 [math.NT], 2021.
%H Hugo van der Sanden, <a href="https://github.com/hvds/seq/tree/master/A101877">Perl and C code</a> for this sequence.
%e The set S = { 1, 2, 3, 6 } gives a sum 1/1 + 1/2 + 1/3 + 1/6 = 2; exhaustive search shows that no set with a smaller maximal element can sum to 2, therefore a(2) = 6.
%e a(1) = 1, S = { 1 }
%e a(2) = 6, S = { 1 2 3 6 }
%e a(3) = 24, S = { 1 2 3 4 5 6 8 9 10 15 18 20 24 }
%e a(4) = 65, S = { 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 }
%e a(5) = 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 }
%e a(6) = 469, using the 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 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 99 100 102 104 105 106 108 110 111 112 114 115 116 117 119 120 121 122 123 124 126 128 129 130 132 133 134 135 136 138 140 141 143 144 145 147 148 150 152 153 154 155 156 159 160 161 162 164 165 168 170 171 174 175 176 177 180 182 183 184 185 186 187 188 189 190 192 195 196 198 200 201 203 204 205 207 208 209 210 212 215 216 217 220 221 222 224 225 228 230 231 232 234 238 240 242 245 246 247 248 250 252 253 255 258 259 260 261 264 266 268 270 272 273 275 276 280 282 285 286 287 288 290 294 295 297 299 300 301 304 305 306 310 312 315 319 320 322 323 324 325 328 329 330 336 340 341 344 345 348 350 351 352 354 357 360 363 364 368 370 372 375 376 377 378 380 384 385 387 390 396 400 402 405 406 407 408 413 414 416 418 424 425 429 430 432 434 435 440 442 444 448 451 455 456 460 462 465 468 469 }
%Y Cf. A002387.
%K nonn,hard,nice,more,changed
%O 1,2
%A _Hugo van der Sanden_, Jan 29 2005
%E Entry revised by _N. J. A. Sloane_, Dec 31 2005
%E Example for a(6) = 469 corrected by _Hugo van der Sanden_, Jan 31 2008
%E a(7)-a(8) from _Hugo van der Sanden_, Jan 20 2015
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