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%I A251637 #21 Dec 17 2014 07:03:20
%S A251637 2,3,4,15,9,8,14,5,15,14,22,35,25,6,6,39,11,7,35,12,12,51,13,33,21,10,
%T A251637 21,16,38,17,26,55,28,20,27,10,69,19,85,65,44,91,45,39,20,87,23,95,34,
%U A251637 91,99,49,85,33,22,62,29,115,57,68,52,77,63,55,45,26,74
%N A251637 Square array read by antidiagonals containing in row n the multiples of prime(n) in A098550 in order of appearance.
%C A251637 T(n,k) = A251715(n,k)*A000040(n); A251715(n,k) = T(n,k)/A000040(n);
%C A251637 T(n,k) = A098550(A251716(n,k)); A251716(n,k) = A098551(T(n,k));
%C A251637 T(n,1) = A251618(n); for n > 4: T(n,2) = A000040(n);
%C A251637 conjecture: A098550 is a permutation of the positive integers iff A001221(n) = number of rows containing n.
%C A251637 A251541 = first column, and A251544 = third column for row numbers > 4. - _Reinhard Zumkeller_, Dec 16 2014
%H A251637 Reinhard Zumkeller, <a href="/A251637/b251637.txt">Rows n = 1..125 of triangle, flattened</a>
%e A251637 . n p | first 14 multiples of p = prime(n) in A098550, n = 1..25
%e A251637 . -------+-------------------------------------------------------------
%e A251637 . 1 2 | 2 4 8 14 6 12 16 10 20 22 26 28 32 18
%e A251637 . 2 3 | 3 9 15 6 12 21 27 39 33 45 51 18 24 36
%e A251637 . 3 5 | 15 5 25 35 10 20 45 85 55 65 30 95 40 50
%e A251637 . 4 7 | 14 35 7 21 28 91 49 63 42 56 77 119 133 161
%e A251637 . 5 11 | 22 11 33 55 44 99 77 66 88 165 143 121 187 110
%e A251637 . 6 13 | 39 13 26 65 91 52 117 78 104 195 143 130 156 221
%e A251637 . 7 17 | 51 17 85 34 68 119 153 102 187 136 170 255 221 204
%e A251637 . 8 19 | 38 19 95 57 133 76 171 114 152 209 247 190 285 228
%e A251637 . 9 23 | 69 23 115 46 161 92 138 207 184 253 299 345 230 276
%e A251637 . 10 29 | 87 29 58 145 203 116 174 261 232 319 377 290 435 348
%e A251637 . 11 31 | 62 31 93 155 124 217 279 186 341 403 248 465 310 372
%e A251637 . 12 37 | 74 37 111 185 148 259 222 333 296 407 555 370 629 481
%e A251637 . 13 41 | 123 41 82 205 164 287 246 369 451 328 410 533 615 492
%e A251637 . 14 43 | 86 43 129 215 172 301 387 258 473 344 430 645 559 516
%e A251637 . 15 47 | 94 47 329 141 235 188 282 423 517 376 470 611 705 564
%e A251637 . 16 53 | 106 53 265 159 212 371 318 477 424 583 689 530 795 636
%e A251637 . 17 59 | 118 59 177 295 236 413 354 531 649 472 767 590 885 1003
%e A251637 . 18 61 | 122 61 427 183 305 244 366 549 671 488 793 610 915 732
%e A251637 . 19 67 | 201 67 335 134 268 469 603 402 536 737 871 670 1005 804
%e A251637 . 20 71 | 142 71 213 355 284 497 426 639 568 781 710 1065 923 852
%e A251637 . 21 73 | 146 73 365 219 292 511 438 657 584 803 730 949 1095 876
%e A251637 . 22 79 | 158 79 237 395 316 553 474 711 632 869 1027 790 1185 948
%e A251637 . 23 83 | 249 83 581 166 415 332 498 747 913 664 1079 830 1245 996
%e A251637 . 24 89 | 178 89 267 445 356 623 534 801 712 979 1157 890 1335 1068
%e A251637 . 25 97 | 291 97 679 194 485 388 582 873 1067 776 970 1261 1455 1164 .
%e A251637 . ---------------------------------------------------------------------
%e A251637 See also A251715 for a table with T(n,k)/p and A251716 for a table of indices of T(n,k) within A098550.
%t A251637 rows = 25; (* f = A098550 *) Clear[f, row]; f[n_ /; n <= 3] := n; f[n_] := f[n] = Module[{k}, For[k=4, GCD[f[n-2], k] == 1 || GCD[f[n-1], k]>1 || MemberQ[Array[f, n-1], k], k++]; k]; row[n_] := row[n] = Module[{k, cnt}, Reap[For[k=1; cnt=0, cnt <= rows-n, k++, If[Divisible[f[k], Prime[n]], cnt++; Sow[f[k]]]]][[2, 1]]]; A251637 = Table[row[n-k+1][[k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Dec 17 2014 *)
%o A251637 (Haskell) when seen as table read by rows:
%o A251637 a251637 n k = a251637_tabl !! (n-1) !! (k-1)
%o A251637 a251637_row n= a251637_tabl !! (n-1)
%o A251637 a251637_tabl = adias $ map
%o A251637 (\k -> filter
%o A251637 ((== 0) . flip mod (fromInteger $ a000040 k)) a098550_list) [1..]
%Y A251637 Cf. A098550, A000040, A251618 (first column), A001221, A251715, A251716.
%Y A251637 Cf. A251541, A251544.
%K A251637 nonn,tabl
%O A251637 1,1
%A A251637 _Reinhard Zumkeller_, Dec 07 2014