A386822 Irregular table T(n,k) = Product_{j = 1..k} prime(j)^(n-j+1), n >= 0, k = 1..n.

1, 2, 4, 12, 8, 72, 360, 16, 432, 10800, 75600, 32, 2592, 324000, 15876000, 174636000, 64, 15552, 9720000, 3333960000, 403409160000, 5244319080000, 128, 93312, 291600000, 700131600000, 931875159600000, 157486901972400000, 2677277333530800000

Offset: 0

Author

Michael De Vlieger, Aug 31 2025

Keywords

• nonn
• tabf
• easy
• new

Comments

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

For n > 1, T(n,n) is in A332785.

For 1 < k < n, T(n,k) is in A286708, where A286708 is the sequence of powerful numbers (i.e., in A001694) that are not prime powers.

For n > 1 and k > 1, T(n,k) is in A126706.

Examples

			Table begins:
  n\k   1      2        3          4           5
  ----------------------------------------------
  0:    1;
  1:    2;
  2:    4,    12;
  3:    8,    72,     360;
  4:   16,   432,   10800,     75600;
  5:   32,  2592,  324000,  15876000,  174636000;
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       1                           .
 2       2 = P(1)^0 * Q(1)   1   0   1
 3       4 = P(1)^1 * Q(1)   1   1   2
 4      12 = P(2)^0 * Q(2)   2   0   2.1
 5       8 = P(1)^2 * Q(1)   1   2   3
 6      72 = P(2)^1 * Q(2)   2   1   3.2
 7     360 = P(3)^0 * Q(3)   3   0   3.2.1
 8      16 = P(1)^3 * Q(1)   1   3   4
 9     432 = P(2)^2 * Q(2)   2   2   4.3
10   10800 = P(3)^1 * Q(3)   3   1   4.3.2
11   75600 = P(4)^0 * Q(4)   4   0   4.3.2.1
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..703 (rows n = 0..37, flattened.)

Crossrefs

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

Programs

Mathematica

Table[Product[Prime[j]^(n - j + 1), {j, k}], {n, 8}, {k, n}] // Flatten

Formula

T(0,1) = 1 by convention.

T(n,1) = A000079(n) = 2^n.

T(n,n) = A006939(n).