P-rough numbers

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):

• 3-rough numbers (  p2-rough numbers): numbers of the form

  2 k + 1, k   ≥   1

  (those are in arithmetic progression);
• 5-rough numbers (  p3-rough numbers): numbers of the form

  6 k ± 1, k   ≥   1

  ;
• 7-rough numbers (  p4-rough numbers): numbers of the form

  30 k ± (1, 7, 11, 13), k   ≥   1

  ;
• 11-rough numbers (  p5-rough numbers): numbers of the form

  210 k ± (1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103), k   ≥   1

  ;
• 13-rough numbers (  p6-rough numbers): numbers of the form

  p5# k ± (1, ...), k   ≥   1

  ;
• 17-rough numbers (  p7-rough numbers): numbers of the form

  p6# k ± (1, ...), k   ≥   1

  ;
• 19-rough numbers (  p8-rough numbers): numbers of the form

  p7# k ± (1, ...), k   ≥   1

  ;
• 23-rough numbers (  p9-rough numbers): numbers of the form

  p8# k ± (1, ...), k   ≥   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

• Smallest prime dividing n
• p -smooth numbers
• An illustration for A002110, A005867, A038110, A060753 covers

  p (2) = 3 ... p (10) = 29

  .