A387493 Numbers of the form P(k)^m * Q(k), k > 1, m >= 1, with P(k) = Product_{i=1..k} prime(i) = A002110(k) and Q(k) = Product_{j=1..k} P(j) = A006939(k).

72, 432, 2592, 10800, 15552, 93312, 324000, 559872, 3359232, 9720000, 15876000, 20155392, 120932352, 291600000, 725594112, 3333960000, 4353564672, 8748000000, 26121388032, 156728328192, 262440000000, 403409160000, 700131600000, 940369969152, 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 >= 1.

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

Proper subset of A286708, which is in turn a proper subset of A001694.

This sequence is A387491 \ A006939 \ A000079 = A387492 \ A006939.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..4100

Example

Table of n, a(n) = P(k)^m * Q(k), for n <= 12, 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         72 = P(2)^1 * Q(2)   2   1   3.2
 2        432 = P(2)^2 * Q(2)   2   2   4.3
 3       2592 = P(2)^3 * Q(2)   2   3   5.4
 4      10800 = P(3)^1 * Q(3)   3   1   4.3.2
 5      15552 = P(2)^4 * Q(2)   2   4   6.5
 6      93312 = P(2)^5 * Q(2)   2   5   7.6
 7     324000 = P(3)^2 * Q(3)   3   2   5.4.3
 8     559872 = P(2)^6 * Q(2)   2   6   8.7
 9    3359232 = P(2)^7 * Q(2)   2   7   9.8
10    9720000 = P(3)^3 * Q(3)   3   3   6.5.4
11   15876000 = P(4)^1 * Q(4)   4   1   5.4.3.2
12   20155392 = P(2)^8 * Q(2)   2   8  10.9

Mathematica

nn = 2^48; k = 2; P = Fold[Times, Prime@ Range[k]]; Q = 2*P;
Union@ Reap[While[j = 1; While[Q*P^j < nn, Sow[Q*P^j]; j++]; j > 1, k++;
  P *= Prime[k]; Q *= P] ][[-1, 1]]

Crossrefs

Cf. A000079, A001694, A002110, A006939, A024619, A025487, A055932, A286708, A387491, A387492.

Sequence in context: A084478 A187158 A164377 * A156632 A353218 A204484

Adjacent sequences: A387490 A387491 A387492 * A387494 A387495 A387496

Keyword

nonn,easy

Author

Michael De Vlieger, Aug 31 2025

Status

approved