|
%I #7 Mar 22 2024 17:39:29
%S 2,4,3,6,9,5,8,10,14,7,12,15,25,22,11,16,20,28,39,26,13,18,21,33,44,
%T 51,34,17,24,27,35,49,52,57,38,19,30,40,55,77,95,68,69,46,23,32,42,56,
%U 78,102,114,76,87,58,29,36,45,65,85,104,115,138,92,93,62
%N Rectangular array read by antidiagonals: row n shows the numbers m >=2 such that the maximum number of consecutive 0's in (e(1), e(2), ..., e(k)) is n-1, where p(1)^e(1) * p(2)^e(2) * ... * p(k)^e(k) is the prime factorization of m.
%C Every positive integer >1 occurs exactly once.
%e Corner:
%e 2 4 6 8 12 16 18 24 30
%e 3 9 10 15 20 21 27 40 42
%e 5 14 25 28 33 35 55 56 65
%e 7 22 39 44 49 77 78 85 88
%e 11 26 51 52 95 102 104 121 143
%e 13 34 57 68 114 115 136 169 171
%e 17 38 69 76 138 145 152 207 217
%e 19 46 87 92 155 174 184 259 261
%e 23 58 93 116 185 186 232 279 287
%e 29 62 111 124 205 222 248 301 333
%e 31 74 123 148 215 246 296 329 369
%e 37 82 129 164 235 258 328 371 387
%e 22 = 2^1 * 3^0 * 5^0 * 7^0 * (11)^1, so (e(1),e(2),e(3),e(4),e(5)) = (1,0,0,0,1), so 22 is in row 4.
%t Map[Transpose[#][[1]] &, GatherBy[Map[{#, Max[Map[Length, DeleteCases[
%t Split[Map[IntegerQ, #/Prime[Range[PrimePi[FactorInteger[#][[-1, 1]]]]]] &[#]], {___, True, ___}]] /. {} -> {0}]} &, Range[2, 400]], #[[2]] &]] // ColumnForm
%t (* _Peter J. C. Moses_, Mar 17 2024 *)
%Y Cf. A000040 (the primes, column 1), A002808 (union of all columns except the first), A055932 (row 1).
%K nonn,tabl
%O 1,1
%A _Clark Kimberling_, Mar 18 2024
|
