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

%I A077011 #34 May 25 2024 15:41:16
%S A077011 1,2,3,6,10,15,30,42,70,105,210,330,462,770,1155,2310,2730,4290,6006,
%T A077011 10010,15015,30030,39270,46410,72930,102102,170170,255255,510510,
%U A077011 570570,746130,881790,1385670,1939938,3233230,4849845,9699690,11741730
%N A077011 Triangle read by rows: T(n,k) = A002110(n)/prime(n+1-k), k = 1..n.
%C A077011 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."
%C A077011 A024451(n) gives the sum of the n-th row.
%C A077011 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 A077011 Alois P. Heinz, <a href="/A077011/b077011.txt">Rows n = 1..130, flattened</a>
%F A077011 A089633(n-1) = Sum_{p | n} 2^(pi(p) - 1) for n > 1, pi(x) = A000720(x). - _Michael De Vlieger_, May 19 2024
%e A077011 Triangle begins:
%e A077011       1;
%e A077011       2,     3;
%e A077011       6,    10,    15;
%e A077011      30,    42,    70,   105;
%e A077011     210,   330,   462,   770,   1155;
%e A077011    2310,  2730,  4290,  6006,  10010,  15015;
%e A077011   30030, 39270, 46410, 72930, 102102, 170170, 255255;
%e A077011   ...
%p A077011 T:= proc(n) local t;
%p A077011       t:= mul(ithprime(i), i=1..n);
%p A077011       seq(t/ithprime(n-i), i=0..n-1)
%p A077011     end:
%p A077011 seq(T(n), n=1..10);  # _Alois P. Heinz_, Jun 04 2012
%t A077011 T[n_] := Module[{t = Product[Prime[i], {i, 1, n}]}, Table[t/Prime[n - i], {i, 0, n - 1}]];
%t A077011 Table[T[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, May 19 2016, translated from Maple *)
%o A077011 (PARI) T(n,k) = vecprod(primes(n))/prime(n+1-k); \\ _Michel Marcus_, May 19 2024
%Y A077011 Row sums give A024451.
%Y A077011 Reversal of A258566.
%Y A077011 Cf. A002110, A089633, A372666.
%K A077011 nonn,tabl
%O A077011 1,2
%A A077011 _Amarnath Murthy_, Oct 26 2002
%E A077011 More terms from _Sascha Kurz_, Jan 26 2003
%E A077011 Name changed by _David James Sycamore_, May 19 2024