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A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).

%I A261144 #77 Feb 16 2025 08:33:26
%S A261144 1,2,1,2,3,6,1,2,3,5,6,10,15,30,1,2,3,5,6,7,10,14,15,21,30,35,42,70,
%T A261144 105,210,1,2,3,5,6,7,10,11,14,15,21,22,30,33,35,42,55,66,70,77,105,
%U A261144 110,154,165,210,231,330,385,462,770,1155,2310,1,2,3,5,6,7,10,11,13,14,15,21,22,26,30,33,35,39,42
%N A261144 Irregular triangle of numbers that are squarefree and smooth (row n contains squarefree p-smooth numbers, where p is the n-th prime).
%C A261144 If we define a triangle whose n-th row consists of all squarefree numbers whose prime factors are all less than prime(k), we get this same triangle except starting with a row {1}, with offset 1. - _Gus Wiseman_, Aug 24 2021
%H A261144 Jean-François Alcover, <a href="/A261144/b261144.txt">Table of n, a(n) for n = 1..2046</a> (first 10 rows)
%H A261144 A. Hildebrand and G. Tenenbaum, <a href="https://eudml.org/doc/93590">Integers without large prime factors</a>, Journal de théorie des nombres de Bordeaux (1993) Volume:5, Issue:2, p. 411-484.
%H A261144 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/SmoothNumber.html">Smooth number</a>.
%H A261144 Wikipedia, <a href="https://en.wikipedia.org/wiki/Smooth_number">Smooth number</a>
%F A261144 T(n-1,k) = A339195(n,k)/prime(n). - _Gus Wiseman_, Aug 24 2021
%e A261144 Triangle begins:
%e A261144 1, 2;                        squarefree and 2-smooth
%e A261144 1, 2, 3, 6;                  squarefree and 3-smooth
%e A261144 1, 2, 3, 5, 6, 10, 15, 30;
%e A261144 1, 2, 3, 5, 6,  7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
%e A261144 ...
%p A261144 b:= proc(n) option remember; `if`(n=0, [1],
%p A261144       sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
%p A261144     end:
%p A261144 T:= n-> b(n)[]:
%p A261144 seq(T(n), n=1..7);  # _Alois P. Heinz_, Nov 28 2015
%t A261144 primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten
%Y A261144 Cf. A000079 (2-smooth), A003586 (3-smooth), A051037 (5-smooth), A002473 (7-smooth), A018336 (7-smooth & squarefree), A051038 (11-smooth), A087005 (11-smooth & squarefree), A080197 (13-smooth), A087006 (13-smooth & squarefree), A087007 (17-smooth & squarefree), A087008 (19-smooth & squarefree).
%Y A261144 Row lengths are A000079.
%Y A261144 Rightmost terms (or column k = 2^n) are A002110.
%Y A261144 Rows are partial unions of rows of A019565.
%Y A261144 Row n is A027750(A002110(n)), i.e., divisors of primorials.
%Y A261144 Row sums are A054640.
%Y A261144 Column k = 2^n-1 is A070826.
%Y A261144 Multiplying row n by prime(n+1) gives A339195, row sums A339360.
%Y A261144 A005117 lists squarefree numbers.
%Y A261144 A056239 adds up prime indices, row sums of A112798.
%Y A261144 A072047 counts prime factors of squarefree numbers.
%Y A261144 A246867 groups squarefree numbers by Heinz weight, row sums A147655.
%Y A261144 A329631 lists prime indices of squarefree numbers, sums A319246.
%Y A261144 A339116 groups squarefree semiprimes by greater factor, sums A339194.
%Y A261144 Cf. A000040, A001221, A006881, A071403, A209862, A319247.
%K A261144 nonn,tabf
%O A261144 1,2
%A A261144 _Jean-François Alcover_, Nov 26 2015