

A261144


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


13



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, 105, 210, 1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 21, 22, 30, 33, 35, 42, 55, 66, 70, 77, 105, 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
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OFFSET

1,2


COMMENTS

If we define a triangle whose nth 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


LINKS

JeanFrançois Alcover, Table of n, a(n) for n = 1..2046 (first 10 rows)
A. Hildebrand, G. Tenenbaum, Integers without large prime factors, Journal de théorie des nombres de Bordeaux (1993) Volume:5, Issue:2, p. 411484.
Eric Weisstein's MathWorld, Smooth number.
Wikipedia, Smooth number


FORMULA

T(n1,k) = A339195(n,k)/prime(n).  Gus Wiseman, Aug 24 2021


EXAMPLE

Triangle begins:
1, 2; squarefree and 2smooth
1, 2, 3, 6; squarefree and 3smooth
1, 2, 3, 5, 6, 10, 15, 30;
1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210;
...


MAPLE

b:= proc(n) option remember; `if`(n=0, [1],
sort(map(x> [x, x*ithprime(n)][], b(n1))))
end:
T:= n> b(n)[]:
seq(T(n), n=1..7); # Alois P. Heinz, Nov 28 2015


MATHEMATICA

primorial[n_] := Times @@ Prime[Range[n]]; row[n_] := Select[ Divisors[ primorial[n]], SquareFreeQ]; Table[row[n], {n, 1, 10}] // Flatten


CROSSREFS

Cf. A000079 (2smooth), A003586 (3smooth), A051037 (5smooth), A002473 (7smooth), A018336 (7smooth & squarefree), A051038 (11smooth), A087005 (11smooth & squarefree), A080197 (13smooth), A087006 (13smooth & squarefree), A087007 (17smooth & squarefree), A087008 (19smooth & squarefree).
Row lengths are A000079.
Rightmost terms (or column k = 2^n) are A002110.
Rows are partial unions of rows of A019565.
Row n is A027750(A002110(n)), i.e., divisors of primorials.
Row sums are A054640.
Column k = 2^n1 is A070826.
Multiplying row n by prime(n+1) gives A339195, row sums A339360.
A005117 lists squarefree numbers.
A056239 adds up prime indices, row sums of A112798.
A072047 counts prime factors of squarefree numbers.
A246867 groups squarefree numbers by Heinz weight, row sums A147655.
A329631 lists prime indices of squarefree numbers, sums A319246.
A339116 groups squarefree semiprimes by greater factor, sums A339194.
Cf. A000040, A001221, A006881, A071403, A209862, A319247.
Sequence in context: A079210 A070861 A277566 * A106524 A323641 A086582
Adjacent sequences: A261141 A261142 A261143 * A261145 A261146 A261147


KEYWORD

nonn,tabf


AUTHOR

JeanFrançois Alcover, Nov 26 2015


STATUS

approved



