

A077011


Triangle in which the nth row contains all possible products of n1 of the first n primes in ascending order.


4



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
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OFFSET

1,2


COMMENTS

A024451(n) gives the sum of the nth 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+1015. 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


LINKS

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


EXAMPLE

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;


MAPLE

T:= proc(n) local t;
t:= mul(ithprime(i), i=1..n);
seq(t/ithprime(ni), i=0..n1)
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 (* JeanFrançois Alcover, May 19 2016, translated from Maple *)


CROSSREFS

Cf. A024451.
Sequence in context: A018141 A178659 A268064 * A246868 A055789 A238891
Adjacent sequences: A077008 A077009 A077010 * A077012 A077013 A077014


KEYWORD

nonn,tabl


AUTHOR

Amarnath Murthy, Oct 26 2002


EXTENSIONS

More terms from Sascha Kurz, Jan 26 2003


STATUS

approved



