A253572 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).

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

Offset

1,3

Comments

Successive rows tend to A000027.

Links

Table of n, a(n) for n=1..78.

Formula

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 ... .

Example

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, ...}

Mathematica

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 *)

Crossrefs

Cf. A000079, A003586, A051037, A002473, A051038 (these are rows 1-5).

Cf. A000027 (natural numbers), A253573.

Sequence in context: A292477 A081532 A174843 * A141539 A340547 A376033

Adjacent sequences: A253569 A253570 A253571 * A253573 A253574 A253575

Keyword

nonn,tabl

Author

L. Edson Jeffery, Jan 03 2015

Extensions

First formula corrected by Tom Edgar, Jan 08 2015

Status

approved