A387839 Numbers of the form P(k)^m * Q(k)^h, k >= 0, m >= 0, h >= 1, with P(k) = Product_{i=1..k} prime(i) = A002110(k) and Q(k) = Product_{j=1..k} P(j) = A006939(k).

1, 2, 4, 8, 12, 16, 32, 64, 72, 128, 144, 256, 360, 432, 512, 864, 1024, 1728, 2048, 2592, 4096, 5184, 8192, 10368, 10800, 15552, 16384, 20736, 31104, 32768, 62208, 65536, 75600, 93312, 124416, 129600, 131072, 186624, 248832, 262144, 324000, 373248, 524288, 559872

Offset

1,2

Comments

Superset of A000079 since P(1)^m * Q(1)^h = 2^(m+h). Intersection of this sequence and A000961 = A000079.

Superset of A006939 since P(k)^0 * Q(k)^1 = A006939(k).

Superset of A368507 = {P(k)^0 * Q(k)^h : h >= 1, k >= 0} (i.e., powers of superprimorials A006939).

Superset of A387491 since P(k)^m * Q(k)^1 is in A387491.

Proper subset of A025487 (i.e., products of primorials), which in turn is a proper subset of A055932 (i.e., numbers with a primorial squarefree kernel).

Squarefree numbers in this sequence are {1, 2}.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Example

Table of n, a(n) for select n showing p-adic valuation of a(n) for p | a(n):
                                  p-adic valuation
 n        a(n)                    2 3 5 7
-----------------------------------------
 1          1 = P(0)^m * Q(0)^h
 2          2 = 2^1               1
 3          4 = 2^2               2
 4          8 = 2^3               3
 5         12 = P(2)^0 * Q(2)^1   2.1
 9         72 = P(2)^1 * Q(2)^1   3.2
11        144 = P(2)^1 * Q(2)^2   4.2
13        360 = P(3)^0 * Q(3)^1   3.2.1
25      10800 = P(3)^1 * Q(3)^1   4.3.2
33      75600 = P(4)^0 * Q(4)^1   4.3.2.1
36     129600 = P(3)^0 * Q(3)^2   6.4.2
41     324000 = P(3)^2 * Q(3)^1   5.4.3
53    3888000 = P(3)^1 * Q(3)^2   7.5.3
61   15876000 = P(4)^1 * Q(4)^1   5.4.3.2

Mathematica

nn = 2^30; k = 1; P = 2; Q = 2;
{1}~Join~Union@ Reap[
  While[i = 1;
    While[j = 0;
      While[Q^i*P^j < nn, Sow[Q^i*P^j]; j++]; j > 0, i++];
      i > 1, k++; P *= Prime[k]; Q *= P] ][[-1, 1]]

Crossrefs

Cf. A000079, A000961, A001694, A006939, A025487, A055932, A126706, A286708, A332785, A368507, A387491.

Sequence in context: A390871 A266047 A375599 * A387491 A324214 A371732

Adjacent sequences: A387836 A387837 A387838 * A387840 A387841 A387842

Keyword

nonn,easy

Author

Michael De Vlieger, Sep 10 2025

Status

approved