A378133 Irregular triangle T(n,k) = P(n)*2^k, n >= 0, k = 0..floor(log_2 prime(k+1)), where P = A002110.

1, 2, 4, 6, 12, 24, 30, 60, 120, 210, 420, 840, 1680, 2310, 4620, 9240, 18480, 30030, 60060, 120120, 240240, 480480, 510510, 1021020, 2042040, 4084080, 8168160, 9699690, 19399380, 38798760, 77597520, 155195040, 223092870, 446185740, 892371480, 1784742960, 3569485920

Offset: 0

Author

Michael De Vlieger, Nov 17 2024

Keywords

• nonn
• tabf
• easy

Comments

Subset of A060735.

a(n) = min(2*a(n-1), f(a(n-1))), where f(k) is the smallest primorial (A002110) greater than k, for n > 0. - Hal M. Switkay, Mar 19 2025

Examples

			Rows n = 0..9:
n\k |         0          1          2           3           4
-------------------------------------------------------------
  0 |         1          .          .           .           .
  1 |         2          4          .           .           .
  2 |         6         12         24           .           .
  3 |        30         60        120           .           .
  4 |       210        420        840        1680           .
  5 |      2310       4620       9240       18480           .
  6 |     30030      60060     120120      240240      480480
  7 |    510510    1021020    2042040     4084080     8168160
  8 |   9699690   19399380   38798760    77597520   155195040
  9 | 223092870  446185740  892371480  1784742960  3569485920
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..3494 (rows n = 0..349, flattened).

Crossrefs

Cf. A000079, A002110, A060735, A088860, A098388, A102476, A378144.

Programs

Mathematica

nn = 16;
MapIndexed[Set[P[First[#2] - 1], #1] &,
  FoldList[Times, 1, Prime@ Range[nn + 1] ] ];
Union@ Flatten@
  Table[P[i]*2^Range[0, Floor[Log2[Prime[i + 1] ] ] ], {i, 0, nn}]

Formula

T(n,k) = A002110(n)*A000079(k), n >= 0, k = 0..A098388(k+1).

T(n,0) = A002110(n).

T(n,1) = A088860(n), n >= 1.

T(n,2) = A102476(n), n >= 2.

T(n,A098388(k+1)) = A378144(n).

Let S(n,j) = A002110(n)*j, n >= 0, j = 0..A006093(n+1) = P(n)*j, n >= 0, j = 0..prime(n+1)-1. Then T(n,k) = S(n, 2^k).