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%I #25 Nov 11 2016 21:21:52
%S 1,1,2,1,2,4,1,2,3,8,1,2,3,4,16,1,2,3,4,6,32,1,2,3,4,5,8,64,1,2,3,4,5,
%T 6,9,128,1,2,3,4,5,6,8,12,256,1,2,3,4,5,6,7,9,16,512,1,2,3,4,5,6,7,8,
%U 10,18,1024,1,2,3,4,5,6,7,8,9,12,24,2048
%N Rectangular array A read by upward antidiagonals in which row A(n) is the sequence of all numbers divisible by no prime exceeding prime(n).
%C Successive rows tend to A000027.
%F A(n) = {prime(1)^(i_1)*...*prime(n)^(i_n) : i_1,...,i_n in {0,1,2,...}}.
%F A(1) subset A(2) subset A(3) subset ... .
%e Array A starts:
%e {1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...}
%e {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, ...}
%e {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, ...}
%e {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, ...}
%e {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, ...}
%e {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...}
%e {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...}
%t r = 20; c = 20; cmax = Max[300, Prime[r + 1]]; a[1] = Table[2^j, {j, 0, cmax}]; b[1] = a[1]; For[n = 2, n <= r, n++, a[n_] := a[n] = {}; b[n_] := b[n] = {}; a[n] = Union[Flatten[Table[Prime[n]^j*b[n - 1], {j, 0, cmax}]]]; For[k = 1, k <= cmax, k++, AppendTo[b[n], a[n][[k]]]]]; Table[b[n - k + 1][[k]], {n, 13}, {k, n}] // Flatten (* Array antidiagonals flattened. *)
%t (* Second program: *)
%t rows = 13; smoothNumbers[p_, max_] := Module[{a, aa, k, pp, iter}, k = PrimePi[p]; aa = Array[a, k]; pp = Prime[Range[k]]; iter = Table[{a[j], 0, PowerExpand @ Log[pp[[j]], max/Times @@ (Take[pp, j-1]^Take[aa, j-1])]}, {j, 1, k}]; Table[Times @@ (pp^aa), Sequence @@ iter // Evaluate] // Flatten // Sort]; t = Table[p = Prime[n]; Take[smoothNumbers[p, If[p == 2, 2^rows, (1/Sqrt[6])* Exp[Sqrt[2*Log[2]*Log[3]*rows]]]], rows-n+1], {n, 1, rows}]; Table[t[[n-k+1, k]], {n, 1, rows}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Nov 09 2016 *)
%Y Cf. A000079, A003586, A051037, A002473, A051038 (these are rows 1-5).
%Y Cf. A000027 (natural numbers), A253573.
%K nonn,tabl
%O 1,3
%A _L. Edson Jeffery_, Jan 03 2015
%E First formula corrected by _Tom Edgar_, Jan 08 2015
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