A362227 a(n) = Product_{k=1..w(n)} p(k)^(S(n,k)-1), where set S(n,k) = row n of A272011 and w(n) = A000120(n) is the binary weight of n.

1, 2, 4, 12, 8, 24, 72, 360, 16, 48, 144, 720, 432, 2160, 10800, 75600, 32, 96, 288, 1440, 864, 4320, 21600, 151200, 2592, 12960, 64800, 453600, 324000, 2268000, 15876000, 174636000, 64, 192, 576, 2880, 1728, 8640, 43200, 302400, 5184, 25920, 129600, 907200, 648000, 4536000, 31752000, 349272000, 15552

Offset: 0

Author

Michael De Vlieger, Jun 08 2023

Keywords

• nonn
• easy

Comments

In other words, let S(n) contain place values of 1's in the binary expansion of n, ordered greatest to least, where S(n,1) = floor(log_2(n+1)) = A000523(n+1) and the remaining terms in S strictly decrease. This sequence reads S(n,k)+1 instead as a multiplicity of prime(k) so as to produce a number with strictly decreasing prime exponents.

Examples

			a(0) = 1 since 1 is the empty product.
a(1) = 2 since 1 = 2^0, s = {0}, hence a(1) = prime(1)^(0+1) = 2^1 = 2.
a(2) = 4 since 2 = 2^1, s = {1}, hence a(2) = 2^(1+1) = 4.
a(3) = 12 since 3 = 2^1+2^0, s = {1,0}, hence a(3) = 2^2*3^1 = 12, etc.
The table below relates first terms of this sequence greater than 1 to A272011 and A067255:
   n   A272011(n)  a(n)  A067255(a(n))
  ------------------------------------
   1   0             2   1
   2   1             4   2
   3   1,0          12   2,1
   4   2             8   3
   5   2,0          24   3,1
   6   2,1          72   3,2
   7   2,1,0       360   3,2,1
   8   3            16   4
   9   3,0          48   4,1
  10   3,1         144   4,2
  11   3,1,0       720   4,2,1
  12   3,2         432   4,3
  13   3,2,0      2160   4,3,1
  14   3,2,1     10800   4,3,2
  15   3,2,1,0   75600   4,3,2,1
  16   4            32   5
  ...
This sequence appears below, seen as an irregular triangle T(m,j) delimited by 2^m where j = 1..2^(m-1) for m > 0:
   1;
   2;
   4, 12;
   8, 24,  72, 360;
  16, 48, 144, 720, 432, 2160, 10800, 75600;
  ...
T(m,1) = 2^m.
T(m,2^(m-1)) = A006939(m) for m > 0.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..16384

Crossrefs

Cf. A000120, A006939, A067255, A087980, A272011.

Programs

Mathematica

Array[Times @@ MapIndexed[Prime[First[#2]]^(#1 + 1) &, Length[#] - Position[#, 1][[All, 1]] ] &[IntegerDigits[#, 2]] &, 48, 0]

Formula

This sequence, sorted, is A087980.

a(2^k) = 2^(k+1).

a(2^k-1) = A006939(k-1).