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

0,2

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.

Links

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

Formula

This sequence, sorted, is A087980.

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

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

Example

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.

Mathematica

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

Crossrefs

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

Sequence in context: A352993 A363250 A181817 * A000348 A168342 A309515

Adjacent sequences: A362224 A362225 A362226 * A362228 A362229 A362230

Keyword

nonn,easy

Author

Michael De Vlieger, Jun 08 2023

Status

approved