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A330394 Irregular triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m such that Omega(m) = n and each prime factor p of m has index pi(p) <= n.

%I A330394 #44 Mar 05 2020 21:05:45
%S A330394 1,2,4,6,9,8,12,18,20,27,30,45,50,75,125,16,24,36,40,54,56,60,81,84,
%T A330394 90,100,126,135,140,150,189,196,210,225,250,294,315,350,375,441,490,
%U A330394 525,625,686,735,875,1029,1225,1715,2401,32,48,72,80,108,112,120,162
%N A330394 Irregular triangle T(n,k) read by rows in which n-th row lists in increasing order all integers m such that Omega(m) = n and each prime factor p of m has index pi(p) <= n.
%C A330394 Positive integers not in T are: 3, 5, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 29, ... .
%C A330394 Row n has exactly one squarefree member: primorial(n) = A002110(n).
%C A330394 Sorting all terms (except 1) gives A324521.
%H A330394 Robert Price, <a href="/A330394/b330394.txt">Table of n, a(n) for n = 0..8788</a>
%e A330394 Triangle T(n,k) begins:
%e A330394   1;
%e A330394   2;
%e A330394   4,  6,  9;
%e A330394   8, 12, 18, 20, 27, 30, 45, 50, 75, 125;
%e A330394   ...
%p A330394 b:= proc(n, i) option remember; `if`(n=0, [1], [seq(
%p A330394       map(x-> x*ithprime(j), b(n-1, j))[], j=1..i)])
%p A330394     end:
%p A330394 T:= n-> sort(b(n$2))[]:
%p A330394 seq(T(n), n=0..5);  # _Alois P. Heinz_, Mar 03 2020
%t A330394 t = Table[Union[Apply[Times, Tuples[Prime[Range[n]], {n}], {1}]], {n, 0, 5}];
%t A330394 t // TableForm
%t A330394 Flatten[t]
%Y A330394 Column k=1 gives A000079.
%Y A330394 Last elements of rows give A307539.
%Y A330394 Row lengths give A088218.
%Y A330394 Row sums give A332967(n) = A124960(2n,n).
%Y A330394 T(n,n) gives A101695(n) for n > 0.
%Y A330394 Cf. A000720, A001222, A005117, A027748, A324521.
%K A330394 nonn,tabf
%O A330394 0,2
%A A330394 _Robert Price_, Mar 03 2020