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A287483 Irregular triangle T(n,k) read by rows: row n lists numbers m with A002110(n) <= m < A002110(n+1) such that omega(m) = n.

%I A287483 #29 Feb 16 2025 08:33:46
%S A287483 1,2,3,5,6,10,14,15,21,22,26,30,42,66,70,78,102,105,110,114,130,138,
%T A287483 154,165,170,174,182,186,190,195,210,330,390,462,510,546,570,690,714,
%U A287483 770,798,858,870,910,930,966,1110,1122,1155,1190,1218,1230,1254,1290
%N A287483 Irregular triangle T(n,k) read by rows: row n lists numbers m with A002110(n) <= m < A002110(n+1) such that omega(m) = n.
%C A287483 The primorial A002110(n) is the smallest squarefree number with n prime factors. Here the n-th row of the triangle is a list of squarefree numbers with n prime factors greater than and including A002110(n) but less than A002110(n+1).
%C A287483 A287484(n) gives row lengths.
%H A287483 Michael De Vlieger, <a href="/A287483/b287483.txt">Table of n, a(n) for n = 0..10309</a> (rows 0 <= n <= 9).
%H A287483 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>
%H A287483 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%e A287483 The sequence begins with 1 as it is equal to A002110(0) and has 0 prime factors. The first primes less than 6 come next, followed by the first squarefree semiprimes (A006881) less than 30 and the smallest terms of A033992 less than 210, etc.
%e A287483 Triangle begins:
%e A287483 n   Row n
%e A287483 0:   1;
%e A287483 1:   2,  3,  5;
%e A287483 2:   6, 10, 14, 15, 21,  22,  26;
%e A287483 3:  30, 42, 66, 70, 78, 102, 105, 110, 114, 130, ..., 195;
%e A287483     ...
%e A287483 In each row n, the squarefree terms m must have omega(m) = n.
%t A287483 Table[Select[Range[#, Prime[n + 1] # - 1] &@ Product[Prime@ i, {i, n}], And[SquareFreeQ@ #, PrimeOmega@ # == n] &], {n, 0, 4}] // Flatten
%Y A287483 Cf. A001221, A002110, A005117, A006881, A033992, A287484, A287691.
%K A287483 nonn,easy,tabf
%O A287483 0,2
%A A287483 _Michael De Vlieger_, May 25 2017
%E A287483 Edited by _N. J. A. Sloane_, Jun 05 2017