A363234 Least number divisible by the first n primes whose factorization into maximal prime powers, if ordered by increasing prime divisor, then has these prime power factors in decreasing order.

1, 2, 12, 720, 151200, 4191264000, 251727315840000, 1542111744113740800000, 10769764221549079560253440000000, 12109394351419848024974600399142912000000000, 78344066654781231654807043124290195568885760000000000, 188552692884723759943358058475004257579791386442930585600000000000

Offset: 0

Author

Michael De Vlieger and David A. Corneth, May 22 2023

Keywords

• nonn
• easy

Comments

a(n) is the least number in A347284 divisible by prime(n).

Also a(n) is the smallest positive integer divisible by prime(n) and prime(i)^e(i) > prime(i + 1)^e(i + 1) where e(k) is the valuation of prime(k) in a(n) and 1 <= i < n. - David A. Corneth, May 24 2023

Equivalently, we can say a(n) is the least number divisible by prime(n) in A363063. This is true also of A363098, the primitive terms of A363063. {a(n)} is the intersection of A347284 and A363098. - Peter Munn, May 29 2023

If we change the end of the sequence name from "decreasing order" to "increasing order", we get the primorial numbers (A002110). - Peter Munn, Jun 04 2023

Examples

			Table shows a(n) = A347284(j) = Product p(i)^m(i), m(i) is the i-th term read from left to right, delimited by ".", in row a(n) of A067255. Example: "4.2.1" signifies 2^4 * 3^2 * 5^1 = 720.
n    j   A067255(a(n))                                   a(n)
-------------------------------------------------------------
0    0                                                      1
1    1   1                                                  2
2    2   2.1                                               12
3    4   4.2.1                                            720
4    5   5.3.2.1                                       151200
5    8   8.5.3.2.1                                 4191264000
6   10   10.6.4.3.2.1                         251727315840000
7   13   13.8.5.4.3.2.1                1542111744113740800000
8   18   18.11.7.5.4.3.2.1   10769764221549079560253440000000
...
		

Links

• Michael De Vlieger, Plot prime(i)^k | a(n) at (x,y) = (k,-n) for n = 1..503.

Crossrefs

Cf. A002110, A067255, A347355.

Subsequence of A347284, A363063, A363098.

Programs

Mathematica

nn = 120; a[0] = {0}; Do[b = {2^k}; Do[If[Last[b] == 1, Break[], i = 1; p = Prime[j]; While[p^i < b[[j - 1]], i++]; AppendTo[b, p^(i - 1)]], {j, 2, k}]; Set[a[k], b], {k, nn}]; s = DeleteCases[Array[a, nn], 1, {2}]; {1}~Join~Table[Times @@ s[[FirstPosition[s, _?(Length[#] == k &)][[1]]]], {k, Max[Length /@ s]}]
(* Generate terms from the linked image. Caution, terms become very large. *)
img = Import["https://oeis.org/A363234/a363234.png", "Image"]; Map[Times @@ MapIndexed[Prime[First[#2]]^#1 &, Reverse@ #] &, SplitBy[Position[ImageData[img][[1 ;; 12]], 0.], First][[All, All, -1]] ]

PARI

a(n) = {resf = matrix(n, 2); resf[,1] = primes(n)~; resf[n, 2] = 1; forstep(j = n-1, 1, -1, resf[j, 2] = logint(resf[j+1, 1]^resf[j+1, 2], resf[j, 1]) + 1); factorback(resf)} \\ David A. Corneth, May 24 2023

Formula

a(n) = A347284(A347355(n)).