A138534 Super least prime signatures; LCM of all signatures with n factors.

1, 2, 12, 120, 5040, 110880, 43243200, 1470268800, 1173274502400, 269853135552000, 516498901446528000, 32022931889684736000, 3234636350177055183360000, 265240180714518525035520000, 1163343432613878250805790720000, 6014485546613750556665938022400000

Offset: 0

Author

Alford Arnold, Mar 28 2008

Keywords

• nonn

Comments

Also the row product of the following table:

1

2

4 3

8 3 5

16 9 5 7

32 9 5 7 11

64 27 25 7 11 13

128 27 25 7 11 13 17

256 81 25 49 11 13 17 19

512 81 125 49 11 13 17 19 23

1024 243 125 49 121 13 17 19 23 29

...

Examples

			For n = 3 the signatures are {8, 12, 30} so a(3) = 120.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..140
• Angelo B. Mingarelli, Abstract factorials, Notes on Number Theory and Discrete Mathematics, Vol. 19, No. 4 (2013), pp. 43-76; arXiv preprint, arXiv:0705.4299 [math.NT], 2007-2012.

Crossrefs

Cf. A006218, A010786, A010786, A010766.

Subsequence of A025487.

LCM of terms in rows of A215366.

Cf. A001222, A006218, A346044.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i<2, 2^n,
      ilcm(seq(b(n-i*j, i-1)*ithprime(i)^j, j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..17);  # Alois P. Heinz, May 15 2015

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i < 2, 2^n, LCM @@ Table[b[n - i j, i - 1] Prime[i]^j, {j, 0, n/i}]];
a[n_] := b[n, n];
a /@ Range[0, 17] (* Jean-François Alcover, Nov 02 2020, after Alois P. Heinz *)
a[n_] := Product[Prime[k]^Floor[n/k], {k, 1, n}]; Array[a, 16, 0] (* Amiram Eldar, Jul 02 2021 *)

PARI

a(n) = prod(k=1, n, prime(k)^(n\k)); \\ Michel Marcus, Jul 03 2021

Formula

From Amiram Eldar, Jul 02 2021: (Start)

a(n) = Product_{k=1..n} prime(k)^floor(n/k).

A001222(a(n)) = A006218(n). (End)

Sum_{n>=0} 1/a(n) = A346044. - Amiram Eldar, Jul 02 2023

Extensions

More terms from Reikku Kulon, Oct 02 2008