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

0,2

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

...

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.

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A006218, A010786, A010786, A010766.

Subsequence of A025487.

LCM of terms in rows of A215366.

Cf. A001222, A006218, A346044.

Sequence in context: A373433 A229901 A007132 * A062080 A221279 A165300

Adjacent sequences: A138531 A138532 A138533 * A138535 A138536 A138537

Keyword

nonn

Author

Alford Arnold, Mar 28 2008

Extensions

More terms from Reikku Kulon, Oct 02 2008

Status

approved