A251637 - OEIS

%I #21 Dec 17 2014 07:03:20

%S 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 21,16,38,17,26,55,28,20,27,10,69,19,85,65,44,91,45,39,20,87,23,95,34,

%U 91,99,49,85,33,22,62,29,115,57,68,52,77,63,55,45,26,74

%N Square array read by antidiagonals containing in row n the multiples of prime(n) in A098550 in order of appearance.

%C T(n,k) = A251715(n,k)*A000040(n); A251715(n,k) = T(n,k)/A000040(n);

%C T(n,k) = A098550(A251716(n,k)); A251716(n,k) = A098551(T(n,k));

%C T(n,1) = A251618(n); for n > 4: T(n,2) = A000040(n);

%C conjecture: A098550 is a permutation of the positive integers iff A001221(n) = number of rows containing n.

%C A251541 = first column, and A251544 = third column for row numbers > 4. - _Reinhard Zumkeller_, Dec 16 2014

%H Reinhard Zumkeller, <a href="/A251637/b251637.txt">Rows n = 1..125 of triangle, flattened</a>

%e .   n   p |  first 14 multiples of p = prime(n) in A098550, n = 1..25

%e .  -------+-------------------------------------------------------------

%e .   1   2 |   2  4   8  14   6  12  16  10   20  22   26   28   32   18

%e .   2   3 |   3  9  15   6  12  21  27  39   33  45   51   18   24   36

%e .   3   5 |  15  5  25  35  10  20  45  85   55  65   30   95   40   50

%e .   4   7 |  14 35   7  21  28  91  49  63   42  56   77  119  133  161

%e .   5  11 |  22 11  33  55  44  99  77  66   88 165  143  121  187  110

%e .   6  13 |  39 13  26  65  91  52 117  78  104 195  143  130  156  221

%e .   7  17 |  51 17  85  34  68 119 153 102  187 136  170  255  221  204

%e .   8  19 |  38 19  95  57 133  76 171 114  152 209  247  190  285  228

%e .   9  23 |  69 23 115  46 161  92 138 207  184 253  299  345  230  276

%e .  10  29 |  87 29  58 145 203 116 174 261  232 319  377  290  435  348

%e .  11  31 |  62 31  93 155 124 217 279 186  341 403  248  465  310  372

%e .  12  37 |  74 37 111 185 148 259 222 333  296 407  555  370  629  481

%e .  13  41 | 123 41  82 205 164 287 246 369  451 328  410  533  615  492

%e .  14  43 |  86 43 129 215 172 301 387 258  473 344  430  645  559  516

%e .  15  47 |  94 47 329 141 235 188 282 423  517 376  470  611  705  564

%e .  16  53 | 106 53 265 159 212 371 318 477  424 583  689  530  795  636

%e .  17  59 | 118 59 177 295 236 413 354 531  649 472  767  590  885 1003

%e .  18  61 | 122 61 427 183 305 244 366 549  671 488  793  610  915  732

%e .  19  67 | 201 67 335 134 268 469 603 402  536 737  871  670 1005  804

%e .  20  71 | 142 71 213 355 284 497 426 639  568 781  710 1065  923  852

%e .  21  73 | 146 73 365 219 292 511 438 657  584 803  730  949 1095  876

%e .  22  79 | 158 79 237 395 316 553 474 711  632 869 1027  790 1185  948

%e .  23  83 | 249 83 581 166 415 332 498 747  913 664 1079  830 1245  996

%e .  24  89 | 178 89 267 445 356 623 534 801  712 979 1157  890 1335 1068

%e .  25  97 | 291 97 679 194 485 388 582 873 1067 776  970 1261 1455 1164 .

%e .  ---------------------------------------------------------------------

%e 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 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 (Haskell) when seen as table read by rows:

%o a251637 n k = a251637_tabl !! (n-1) !! (k-1)

%o a251637_row n= a251637_tabl !! (n-1)

%o a251637_tabl = adias $ map

%o    (\k -> filter

%o      ((== 0) . flip mod (fromInteger $ a000040 k)) a098550_list) [1..]

%Y Cf. A098550, A000040, A251618 (first column), A001221, A251715, A251716.

%Y Cf. A251541, A251544.

%K nonn,tabl

%O 1,1

%A _Reinhard Zumkeller_, Dec 07 2014