Value a(6) in doubt for a = A101877

Tue Jan 15 18:21:31 CET 2008

so it is similarly possible that multiple solutions of different
solution: S_7 = {
  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 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81
  82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105
  106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124
  125 126 127 128 129 130 131 132 133 134 135 136 137 138 140 141 142 143 144
  145 146 147 148 150 152 153 154 155 156 158 159 160 161 162 164 165 166 168
  169 170 171 172 173 174 175 176 177 178 180 182 183 184 185 186 187 188 189
  190 192 194 195 196 198 200 201 202 203 204 205 206 207 208 209 210 212 213
  214 215 216 217 218 219 220 221 222 224 225 226 228 230 231 232 234 235 236
  237 238 240 242 243 244 245 246 247 248 249 250 252 253 254 255 256 258 259
  260 261 262 264 265 266 267 268 270 272 273 274 275 276 278 279 280 282 284
  285 286 287 288 290 291 294 295 296 297 298 299 300 301 303 304 305 306 308
  309 310 312 315 316 318 319 320 321 322 323 324 325 328 329 330 332 333 335
  336 338 339 340 341 342 344 345 346 348 350 351 352 354 355 356 357 360 363
  364 365 366 368 369 370 371 372 374 375 376 377 378 380 381 384 385 387 388
  390 391 392 393 395 396 399 400 402 403 404 405 406 407 408 410 411 412 413
  414 415 416 417 418 420 423 424 425 426 427 428 429 430 432 434 435 436 437
  438 440 441 442 444 445 448 450 451 452 455 456 459 460 462 464 465 468 469
  470 472 473 474 475 476 477 480 481 483 484 485 486 488 490 492 493 494 495
  496 497 498 500 504 505 506 510 511 513 516 517 518 520 522 525 527 528 530
  531 532 533 534 535 536 539 540 544 545 546 548 549 550 551 552 553 555 558
  559 560 561 564 565 567 568 570 572 574 575 576 580 581 582 583 584 585 588
  589 590 592 594 595 596 598 600 605 608 609 611 612 615 616 618 620 621 624
  627 629 630 632 636 637 638 639 640 642 644 645 646 648 649 650 651 654 655
  656 657 658 660 663 664 665 666 667 670 671 672 675 676 679 680 684 685 688
  689 690 692 693 695 696 697 700 702 703 704 705 707 708 710 711 712 713 715
  720 721 725 726 728 730 731 732 735 736 737 738 740 741 742 744 745 747 748
  750 752 754 756 759 760 762 763 765 767 768 770 774 775 776 777 780 781 782
  783 784 791 792 793 795 798 799 800 801 803 804 806 808 810 812 814 816 817
  819 820 824 825 826 828 832 833 836 837 840 845 846 847 848 850 851 852 854
  855 856 860 861 864 868 869 870 871 873 874 875 876 880 882 884 885 888 889
  891 893 894 896 897 899 900 901 902 903 909 910 912 913 915 917 918 920 923
  924 925 928 930 931 935 936 938 943 944 945 946 949 950 952 954 957 960 962
  966 968 969 970 972 975 976 979 980 981 984 986 987 988 989 990 992 994 996
  999 1000 1001 1005 1007 1008 1010 1012 1014 1016 1020 1022 1023 1025 1026
  1030 1032 1034 1035 1036 1037 1038 1040 1044 1045 1050 1053 1054 1056 1060
  1064 1066 1067 1068 1070 1071 1072 1075 1078 1080 1081 1085 1088 1089 1092
  1095 1100 1104 1105 1106 1107 1110 1112 1113 1116 1118 1120 1121 1122 1125
  1127 1128 1130 1131 1133 1136 1139 1140 1144 1147 1148 1150 1155 1157 1159
  1160 1161 1166 1168 1170 1173 1175 1178 1179 1180 1183 1185 1188 1189 1190
  1192 1196 1199 1200 1204 1206 1209 1210 1211 1212 1215 1218 1220 1221 1224
  1225 1230 1232 1240 1241 1242 1243
since 1/4 = 1/10 + 1/12 + 1/15, which nicely demonstrates my second caveat.
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Message-ID: <3c3af2330801151034k4b6f1f6ve08f3e8a34632578 at mail.gmail.com>
Date: Tue, 15 Jan 2008 14:34:24 -0400
From: "Maximilian Hasler" <maximilian.hasler at gmail.com>
To: "David W. Wilson" <wilson.d at anseri.com>
Subject: Re: formula for A103314(2pq)
Cc: seqfan at ext.jussieu.fr
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(I just noticed that you sent a copy to seqfan, so I send also a
summary of my reply in order to save others the time of writing
dupilcate replies)

On Jan 15, 2008 11:23 AM, David W. Wilson <wilson.d at anseri.com> wrote:
> > after inspection, I conjecture that none of both is needed :
> > it is sufficient to start with {} and { e_n^k U_d , d|n, d>1 }
> > and repeatedly add disjoint unions (e_n=exp(i2pi/n)).

(This applies to the semiprime case you referred to in your message ;
as I wrote, I agree difference is needed for more than 2 prime factors.)

> Your base set { e_n^k U_d , d|n, d>1 } tacitly includes rotations (...)
> but you have not dispense with the rotations from a computational or
> mathematical standpoint.
> And yes, I recognize that your base set {{}} U { U_d: d | n } is a little
> different from my { U_p : p prime, p | n }, but with a little work, I can
> show that all the elements of your base set are generated from mine,

The point is, by starting with a base set which is rotationally
invariant, you don't need to apply further rotations to the
additionnally constructed sets.

Of course I agree that starting with all d|n, d>1, is not necessary,
but gives all sets you get by taking unions of differently rotated
U_p, without the need of checking whether the union is allowed and/or
has already been added to the collection.

> > PS: no problem if you can't run my code, it was just for illustration
> > of my algorithm (yours simplified) and what I told (storing subsets as
> > sum(2^i), doing intersection through bitand(), etc)
> As a programming geek, my approach would
> be to model these sets as bit vectors (integers) and use shifts and logical
> bit operations to effect rotations, complements and unions. I'm guessing

(Thats what I do and intended to illustrate...)

M.H.