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%I #22 Feb 08 2024 17:41:37
%S 1,2,3,6,10,15,30,42,70,105,210,330,462,770,1155,2310,2730,4290,6006,
%T 10010,15015,30030,39270,46410,72930,102102,170170,255255,510510,
%U 570570,746130,881790,1385670,1939938,3233230,4849845,9699690,11741730
%N Triangle in which the n-th row contains all possible products of n-1 of the first n primes in ascending order.
%C A024451(n) gives the sum of the n-th row.
%C 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
%H Alois P. Heinz, <a href="/A077011/b077011.txt">Rows n = 1..130, flattened</a>
%e Triangle begins:
%e 1;
%e 2, 3;
%e 6, 10, 15;
%e 30, 42, 70, 105;
%e 210, 330, 462, 770, 1155;
%e 2310, 2730, 4290, 6006, 10010, 15015;
%e 30030, 39270, 46410, 72930, 102102, 170170, 255255;
%e ...
%p T:= proc(n) local t;
%p t:= mul(ithprime(i), i=1..n);
%p seq(t/ithprime(n-i), i=0..n-1)
%p end:
%p seq(T(n), n=1..10); # _Alois P. Heinz_, Jun 04 2012
%t T[n_] := Module[{t = Product[Prime[i], {i, 1, n}]}, Table[t/Prime[n - i], {i, 0, n - 1}]];
%t Table[T[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, May 19 2016, translated from Maple *)
%Y Row sums give A024451.
%Y Reversal of A258566.
%K nonn,tabl
%O 1,2
%A _Amarnath Murthy_, Oct 26 2002
%E More terms from _Sascha Kurz_, Jan 26 2003
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