A079474 Triangular array: for s=0 to r-1, a(r,s) = p(s)^(r-s), where p(s) is the s-th primorial number. (p(0)=1, p(1)=2, p(2)=2*3, p(3)=2*3*5,...).

1, 1, 2, 1, 4, 6, 1, 8, 36, 30, 1, 16, 216, 900, 210, 1, 32, 1296, 27000, 44100, 2310, 1, 64, 7776, 810000, 9261000, 5336100, 30030, 1, 128, 46656, 24300000, 1944810000, 12326391000, 901800900, 510510, 1, 256, 279936, 729000000, 408410100000

Offset: 1

Author

Alford Arnold, Jan 15 2003

Keywords

• nonn
• tabl

Comments

In the expansion of [1+x+x^2+...+x^(r-s)]^s, the x^n coefficient states how many factors of a(r,s) have n prime factors.

As a square array A(n,k) n>=0 k>=1 read by descending antidiagonals, A(n,k) when n>=1 is the least common period over the positive integers of the occurrence of the first n prime numbers as the k-th least operand in the respective integers' prime factorizations (written without exponents). - Peter Munn, Jan 25 2017

Examples

			Triangle starts
  1;
  1,  2;
  1,  4,    6;
  1,  8,   36,    30;
  1, 16,  216,   900,   210;
  1, 32, 1296, 27000, 44100, 2310;
  ...
		

Crossrefs

Cf. A002110, A007318, A008287, A027907, A036035, A061742, A074139.

a(2n,n) gives A181555.

Programs

Maple

p:= proc(n) option remember; `if`(n=0, 1, ithprime(n)*p(n-1)) end:
a:= (r, s)-> p(s)^(r-s):
seq(seq(a(r, s), s=0..r-1), r=0..10);  # Alois P. Heinz, Aug 22 2019

Mathematica

p[0] = 1; p[s_] := p[s] = Prime[s] p[s-1];
a[r_, s_] := p[s]^(r-s);
Table[a[r, s], {r, 0, 10}, {s, 0, r-1}] // Flatten (* Jean-François Alcover, Dec 07 2019 *)

Extensions

Edited by Don Reble, Nov 02 2005