OFFSET
0,2
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
LINKS
Michael De Vlieger, Plot prime(i)^k | a(n) at (x,y) = (k,-n) for n = 1..503.
EXAMPLE
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
...
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]] ]
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger and David A. Corneth, May 22 2023
STATUS
approved