A087005 -id:A087005 - cp's OEIS frontend

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

Jean-François Alcover, Nov 26 2015

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

			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;
...

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

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.

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

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

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

A087006 Divisors of 30030.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 21, 22, 26, 30, 33, 35, 39, 42, 55, 65, 66, 70, 77, 78, 91, 105, 110, 130, 143, 154, 165, 182, 195, 210, 231, 273, 286, 330, 385, 390, 429, 455, 462, 546, 715, 770, 858, 910, 1001, 1155, 1365, 1430, 2002, 2145, 2310, 2730

Offset: 1

Reinhard Zumkeller, Jul 29 2003

30030 = 13# = A002110(6);
divisors of 30030 are squarefree (A005117) and 13-smooth;
a(A000005(30030)) = a(64) = 30030 is the last term.

Nathaniel Johnston, Table of n, a(n) for n = 1..64 (full sequence)
Index entries for sequences related to divisors of numbers
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Divisor

Cf. A018255, A018336, A087005, A087007, A087008.

a=30030;Divisors[a] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2009 *)

PARI
```
divisors(30030)
```

A087007 Divisors of 510510.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 21, 22, 26, 30, 33, 34, 35, 39, 42, 51, 55, 65, 66, 70, 77, 78, 85, 91, 102, 105, 110, 119, 130, 143, 154, 165, 170, 182, 187, 195, 210, 221, 231, 238, 255, 273, 286, 330, 357, 374, 385, 390, 429, 442, 455, 462, 510, 546, 561

Offset: 1

Reinhard Zumkeller, Jul 29 2003

510510 = 17# = A002110(7);
divisors of 510510 are squarefree (A005117) and 17-smooth;
a(A000005(510510)) = a(128) = 510510 is the last term.

R. Zumkeller, Table of n, a(n) for n = 1..128
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Divisor
Index entries for sequences related to divisors of numbers

Cf. A018255, A018336, A087005, A087006, A087008.

a=510510;Divisors[a] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2009 *)

PARI
```
divisors(510510)
```

A087008 Divisors of 9699690.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 51, 55, 57, 65, 66, 70, 77, 78, 85, 91, 95, 102, 105, 110, 114, 119, 130, 133, 143, 154, 165, 170, 182, 187, 190, 195, 209, 210, 221, 231, 238, 247, 255, 266, 273, 285, 286, 323, 330

Offset: 1

Reinhard Zumkeller, Jul 29 2003

9699690 = 19# = A002110(8);
divisors of 9699690 are squarefree (A005117) and 19-smooth;
a(A000005(9699690)) = a(256) = 9699690 is the last term.

R. Zumkeller, Table of n, a(n) for n = 1..256
Eric Weisstein's World of Mathematics, Primorial
Eric Weisstein's World of Mathematics, Divisor
Index entries for sequences related to divisors of numbers

Cf. A018255, A018336, A087005, A087006, A087007.

a=9699690;Divisors[a] (* Vladimir Joseph Stephan Orlovsky, Jan 25 2009 *)

divisors(9699690) \\ Charles R Greathouse IV, Apr 25 2011

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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).

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A087006 Divisors of 30030.

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A087007 Divisors of 510510.

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A087008 Divisors of 9699690.

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