A077011 Triangle read by rows: T(n,k) = A002110(n)/prime(n+1-k), k = 1..n.

1, 2, 3, 6, 10, 15, 30, 42, 70, 105, 210, 330, 462, 770, 1155, 2310, 2730, 4290, 6006, 10010, 15015, 30030, 39270, 46410, 72930, 102102, 170170, 255255, 510510, 570570, 746130, 881790, 1385670, 1939938, 3233230, 4849845, 9699690, 11741730

Offset: 1

Author

Amarnath Murthy, Oct 26 2002

Keywords

• nonn
• tabl

Comments

Original Name was: "Triangle in which the n-th row contains all possible products of n-1 of the first n primes in ascending order."

A024451(n) gives the sum of the n-th row.

When the triangle is parsed in blocks of ascending length, as shown in the example, there is the following interpretation: The integers Z regarded as a module over themselves contain unshortenable generating sets of different lengths, in fact, infinitely many of each desired length. Each of the blocks is the minimal example of an unshortenable generating set of the respective length. For example, {6,10,15} generates Z as 1=6+10-15. However, removing one of the numbers leaves two numbers that are not relatively prime, precluding generation of Z. An analogous argument succeeds for all other blocks alike. Each block contains numbers such that there is no prime factor common to all. Taking differences sufficiently often one ends up with two coprime numbers whence the generating property follows from Bézout's theorem. If just one number is removed from the set, relative primality is lost. The minimality of the numbers used in each block is evident from the construction. - Peter C. Heinig (algorithms(AT)gmx.de), Oct 04 2006

Examples

			Triangle begins:
      1;
      2,     3;
      6,    10,    15;
     30,    42,    70,   105;
    210,   330,   462,   770,   1155;
   2310,  2730,  4290,  6006,  10010,  15015;
  30030, 39270, 46410, 72930, 102102, 170170, 255255;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..130, flattened

Crossrefs

Row sums give A024451.

Reversal of A258566.

Cf. A002110, A089633, A372666.

Programs

Maple

T:= proc(n) local t;
      t:= mul(ithprime(i), i=1..n);
      seq(t/ithprime(n-i), i=0..n-1)
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Jun 04 2012

Mathematica

T[n_] := Module[{t = Product[Prime[i], {i, 1, n}]}, Table[t/Prime[n - i], {i, 0, n - 1}]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, May 19 2016, translated from Maple *)

PARI

T(n,k) = vecprod(primes(n))/prime(n+1-k); \\ Michel Marcus, May 19 2024

Formula

A089633(n-1) = Sum_{p | n} 2^(pi(p) - 1) for n > 1, pi(x) = A000720(x). - Michael De Vlieger, May 19 2024

Extensions

More terms from Sascha Kurz, Jan 26 2003

Name changed by David James Sycamore, May 19 2024