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P-rough numbers

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A
p
-rough number
, or
p
-jagged number
, is an integer whose smallest prime factor is
p
(Finch, 2001).

The 2-rough numbers (  p1-rough numbers) are the integers excluding zero and the units −1 and +1. The 3-rough numbers are the odd numbers excluding the units −1 and +1.

Among the positive integers (greater than 1):

where
pn#
is the
n
-th primorial number.
p
p
-rough numbers*
A-number
2 {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, ...} A000027
 (n), n   ≥   2
3 {3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, ...} A005408
 (n), n   ≥   2
5 {5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49, 53, 55, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 121, 125, ...} A007310
 (n), n   ≥   2
7 {7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 91, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 143, 149, ...} A007775
 (n), n   ≥   2
11 {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, ...} A008364
 (n), n   ≥   2
13 {13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, ...} A008365
 (n), n   ≥   2
17 {17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, ...} A008366
 (n), n   ≥   2
19 {19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, ...} A166061
 (n), n   ≥   2
23 {23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, ...} A166063
 (n), n   ≥   2
29 {29, 31, 37, 41, 43, 47, ...}  
* The OEIS considers 1 as a
p
-rough number for any
p
, although it doesn’t have a smallest prime factor (it is the empty product of primes).

Density of prime (n)-rough numbers

The density of prime (n)-rough numbers is

   
{1, 1 −
1
2
 , 1 −
1
2
1
6
 , 1 −
1
2
1
6
1
15
 , 1 −
1
2
1
6
1
15
4
105
 , 1 −
1
2
1
6
1
15
4
105
8
385
 , ...} =
{1,
1
2
 ,
1
3
 ,
4
15
 ,
8
35
 ,
16
77
 ,
192
1001
 ,
3072
17017
 ,
55296
323323
 , ...}

where

   
{
1
2
 ,
1
6
 ,
1
15
 ,
4
105
 ,
8
385
 ,
16
1001
 ,
192
17017
 ,
3072
323323
 ,
55296
7436429
 , ...} =
{1 −
1
2
 ,
1
2
1
3
 ,
1
3
4
15
 ,
4
15
8
35
 , ...}

is the density of positive integers with smallest prime factor prime (n), which is equal to the density of prime (n)-rough numbers minus the density of prime (n + 1)-rough numbers.

A038110 Numerator of
n  − 1

k  = 1
(1  − 
1
prime (k )
  )
. Numerator of density of integers with smallest prime factor prime (n). Numerator of density of prime (n)-rough numbers.
{1, 1, 1, 4, 8, 16, 192, 3072, 55296, 110592, 442368, 13271040, 477757440, 19110297600, 802632499200, 1605264998400, 6421059993600, 12842119987200, 770527199232000, 50854795149312000, 3559835660451840000, ...}
A060753 Denominator of
n  − 1

k  = 1
(1  − 
1
prime (k )
  )
. Denominator of density of prime (n)-rough numbers.
{1, 2, 3, 15, 35, 77, 1001, 17017, 323323, 676039, 2800733, 86822723, 3212440751, 131710070791, 5663533044013, 11573306655157, 47183480978717, 95993978542907, 5855632691117327, 392327390304860909, ...}
A038111 Denominator of
1
prime (n)
n  − 1

k  = 1
(1  − 
1
prime (k )
  )
. Denominator of density of integers with smallest prime factor prime (n).
     
{2, 6, 15, 105, 385, 1001, 17017, 323323, 7436429, 19605131, 86822723, 3212440751, 131710070791, 5663533044013, 266186053068611, 613385252723321, 2783825377744303, 5855632691117327, 392327390304860909, ...}

Sequences

A063538 Not
2  n  −  1
-smooth numbers: largest prime factor of
n
(A006530)
  ≥  
2  n
. (Thus
2  n
-rough numbers.)
{2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, ...}

See also