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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). 3
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A243851 A168266

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

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Last modified February 23 18:48 EST 2019. Contains 320438 sequences. (Running on oeis4.)