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A253572
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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).
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3
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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, 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, 10, 18, 1024, 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 2048
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OFFSET
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1,3
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COMMENTS
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LINKS
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FORMULA
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A(n) = {prime(1)^(i_1)*...*prime(n)^(i_n) : i_1,...,i_n in {0,1,2,...}}.
A(1) subset A(2) subset A(3) subset ... .
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EXAMPLE
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Array A starts:
{1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ...}
{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, ...}
{1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, ...}
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, ...}
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, ...}
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...}
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, ...}
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MATHEMATICA
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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. *)
(* Second program: *)
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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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First formula corrected by Tom Edgar, Jan 08 2015
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STATUS
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approved
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