A347284 - OEIS

%I #18 May 19 2023 17:26:51

%S 1,2,12,24,720,151200,302400,1814400,4191264000,8382528000,

%T 251727315840000,503454631680000,3020727790080000,

%U 1542111744113740800000,3084223488227481600000,92526704646824448000000,555160227880946688000000,1110320455761893376000000,10769764221549079560253440000000

%N a(n) = Product_{j=1..A089576(n)} p_j^e_j with e_j = floor(e_(j-1)*log(p_(j-1))/log(p_j)) where the first factor is 2^n.

%C a(n) is the product of the largest prime power divisors p_j^e_j such that p_j^e_j < p_(j-1)^e_(j-1), beginning with p_1^e_1 = 2^n and proceeding with the next prime p until e_j = 0.

%C {a(n)} is a subset of A025487 which is a subset of A055932. All terms are products of primorials. No primes p_j for 1 <= j <= L have e = 0 with the exception of a(0) = 2^0. Let L = A001221(a(n)).

%C The largest primorial divisor P(L) = A2110(L).

%C For n > 0, all terms are even.

%C The greatest prime divisor p_L has multiplicity e_L = 1.

%C All multiplicities e are distinct; for 1 <= j <= L, the multiplicity e_j >= L - j + 1.

%C a(k) | a(n) for 0 <= k <= n.

%C The numbers q = a(n+1)/a(n) are primorials.

%C Finite intersection of A002182 and a(n) = {1, 2, 12, 360, 75600}.

%C Chernoff number A006939(L) | a(n). Quotient K = a(n) | A006939(L) is in A025487.

%C The prime shape of terms resembles a simplified map of the US state of Idaho.

%H Michael De Vlieger, <a href="/A347284/b347284.txt">Table of n, a(n) for n = 0..144</a>

%H Michael De Vlieger, <a href="/A347284/a347284.png">Bitmap resulting from binary compactification of a(n)</a>, 0 <= n <= 4096.

%H Michael De Vlieger, <a href="/A347284/a347284.gif">Animation of prime shapes of a(n)</a> for 2 <= n <= 37, illustrating a(n) as a product of a particular sequence of primorials.

%F a(n) = Product_{j=1..k} p_j^T(n,j) where T = A347285 and k = A089576(n).

%F Row n of A347285 yields row a(n) of A067255.

%F a(n) = product of row n of A347288.

%e a(0) = 2^0 = 1;

%e a(1) = 2^1 = 2, since 3^1 > 2^1;

%e a(2) = 2^2 * 3^1, since 3^1 < 2^2 but 3^2 > 2^2, and since 5^1 > 3^1;

%e a(3) = 2^3 * 3^1, since 3^1 < 2^3 but 3^2 > 2^3, and 5^1 > 3^1;

%e a(4) = 2^4 * 3^2 * 5^1, since 3^2 < 2^4 yet 3^3 > 2^4, 5^1 < 3^2 yet 5^2 > 3^2, and 7^1 > 5^1; etc.

%e Prime shapes of a(n) for 2 <= n <= 5:

%e                                                      5  o

%e                                     4  o             4  x

%e                      3  o           3  x             3  x x

%e       2  x           2  x           2  x x           2  x x x

%e a(2)  1  X X   a(3)  1  X X   a(4)  1  X X X   a(5)  1  X X X X

%e          2 3            2 3            2 3 5            2 3 5 7

%e This demonstrates that a(n) is in A025487, that A002110(A001221(a(n))) is the greatest primorial divisor of a(n) as a consequence (prime divisors represented by capital X's), and Chernoff A006939(A001221(a(n))) | n, prime divisors represented by x's of any case. a(n) = A006939(A001221(a(n))) * k, k in A025487, represented by o's.

%e Because each multiplicity e is necessarily distinct, we may compactify a(n) using Sum_{k=1..omega(a(n))} 2^(e-1).

%e Prime shapes of a(12):

%e       12  o

%e       11  o

%e       10  o

%e        9  o

%e        8  o

%e        7  o o

%e        6  x o

%e        5  x x

%e        4  x x x

%e        3  x x x x

%e        2  x x x x x

%e a(12)  1  X X X X X X

%e           2 3 5 7 ...

%e a(12) = A006939(6) * 2^6 * 3^2

%e       = 5244319080000 * 64 * 9

%e       = 3020727790080000.

%e                                                  O

%e                                        O         x

%e                             O          x         x

%e                   O         x          x o       x x

%e          O        x         x o        x x o     x x x

%e   O      x o      x x       x x o      x x x o   x x x x

%e a(1)*6 = a(2)*2 = a(3)*30 = a(4)*210 = a(5)*2 =  a(6), etc., hence a(n) can be generated by a list of indices of primorials {1, 2, 1, 3, 4, 1, 1, 5, ...} and thereby be efficiently compactified.

%t Array[Times @@ NestWhile[Append[#1, #2^Floor@ Log[#2, #1[[-1]]]] & @@ {#, Prime[Length@ # + 1]} &, {2^#}, Last[#] > 1 &] &, 18, 0] (* or *)

%t Block[{nn = 2^5, a = {}, b, e, i, m, p}, Array[Set[e[#], 0] &, Floor[2^# If[# <= 4, 1/2, -1 + 2^(7/(3 #))]] &[Ceiling@ Log2@ nn]]; Do[e[1]++; b = {2^e[1]}; Do[If[Last[b] == 1, Break[], i = e[j]; p = Prime[j]; While[p^i < b[[j - 1]], i++]; AppendTo[b, p^(i - 1)]; If[i > e[j], e[j]++]], {j, 2, k}]; AppendTo[a, Times @@ b], {k, nn}]; Prepend[a, 1]]

%t (* Generate up to 4096 terms from the bitmap image *)

%t With[{r = ImageData@ Import["https://oeis.org/A347284/a347284.png"]}, {1}~Join~Table[Times @@ Flatten@ MapIndexed[Prime[#2]^#1 &, Reverse@ Position[r[[i]], 0.][[All, 1]]], {i, 20}]]

%t (* Generate up to 10000 terms using b-file at A347354 (numbers are large as n increases, limit nn is set to 120): *)

%t Block[{nn = 120, s, m}, s = Import["https://oeis.org/A347354/b347354.txt", "Data"][[1 ;; nn, -1]]; m = Prime@ Range@ Max[s]; {1}~Join~FoldList[Times, Map[Times @@ m[[1 ;; #]] &, s]]] (* _Michael De Vlieger_, Sep 25 2021 *)

%Y Cf. A000079, A001221, A002110, A002182, A006939, A067255, A089576, A347285, A347288, A347354.

%Y Subsequence of A025487, A055932, A363063.

%K nonn

%O 0,2

%A _Michael De Vlieger_ and _David James Sycamore_, Aug 26 2021

%E Definition edited by _Peter Munn_, May 19 2023