A387492 Numbers of the form P(k)^m * Q(k), k > 1, m >= 0, with P(k) = Product_{i=1..k} prime(i) = A002110(k) and Q(k) = Product_{j=1..k} P(j) = A006939(k).

12, 72, 360, 432, 2592, 10800, 15552, 75600, 93312, 324000, 559872, 3359232, 9720000, 15876000, 20155392, 120932352, 174636000, 291600000, 725594112, 3333960000, 4353564672, 8748000000, 26121388032, 156728328192, 262440000000, 403409160000, 700131600000, 940369969152, 5244319080000, 5642219814912, 7873200000000, 33853318889472, 147027636000000, 203119913336832, 236196000000000

Offset

1,1

Comments

Numbers that are products of a power of primorial P(k)^m and a superprimorial Q(k), k > 1, m >= 0.

Proper subset of A025487, which in turn is a proper subset of A055932.

Superset of A006939.

This sequence is A387491 \ A000079.

Intersection of this sequence and A286708 is A387493.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..4136

Example

Table of n, a(n) = P(k)^m * Q(k), for n <= 14, illustrating prime power factor exponents, where k = omega(a(n)) = A001221(a(n)), P = A002110, and Q = A006939:
                                          Exponents of
 n          a(n)                  k   m   2.3.5.7
------------------------------------------------------
 1           12 = P(2)^0 * Q(2)   2   0   2.1
 2           72 = P(2)^1 * Q(2)   2   1   3.2
 3          360 = P(3)^0 * Q(3)   3   0   3.2.1
 4          432 = P(2)^2 * Q(2)   2   2   4.3
 5         2592 = P(2)^3 * Q(2)   2   3   5.4
 6        10800 = P(3)^2 * Q(3)   3   2   4.3.2
 7        15552 = P(2)^4 * Q(2)   2   4   6.5
 8        75600 = P(4)^0 * Q(4)   4   0   4.3.2.1
 9        93312 = P(2)^5 * Q(2)   2   5   7.6
10       324000 = P(3)^2 * Q(3)   4   2   5.4.3
11       559872 = P(2)^6 * Q(2)   2   6   8.7
12      3359232 = P(2)^7 * Q(2)   2   7   9.8
13      9720000 = P(3)^3 * Q(3)   3   3   6.5.4
14     15876000 = P(4)^1 * Q(4)   4   1   5.4.3.2

Mathematica

nn = 2^40; k = 2; P = Fold[Times, Prime@ Range[k]]; Q = 2*P;
Union@ Reap[While[j = 0; While[Q*P^j < nn, Sow[Q*P^j]; j++]; j > 0, k++;
  P *= Prime[k]; Q *= P] ][[-1, 1]]

Crossrefs

Cf. A000079, A002110, A006939, A024619, A025487, A055932, A387491, A387493.

Sequence in context: A227022 A036392 A035472 * A036398 A125322 A126480

Adjacent sequences: A387489 A387490 A387491 * A387493 A387494 A387495

Keyword

nonn,easy

Author

Michael De Vlieger, Aug 31 2025

Status

approved