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

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

Offset: 1

Author

Jean-François Alcover, Nov 26 2015

Keywords

• nonn
• tabf

Comments

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

Examples

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

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..2046 (first 10 rows)
• A. Hildebrand and G. Tenenbaum, Integers without large prime factors, Journal de théorie des nombres de Bordeaux (1993) Volume:5, Issue:2, p. 411-484.
• Eric Weisstein's MathWorld, Smooth number.
• Wikipedia, Smooth number

Crossrefs

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).

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^n-1 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.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, [1],
      sort(map(x-> [x, x*ithprime(n)][], b(n-1))))
    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

Formula

T(n-1,k) = A339195(n,k)/prime(n). - Gus Wiseman, Aug 24 2021