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P-rough numbers
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(Redirected from 29-rough numbers)
A
-rough number, or
-jagged number, is an integer whose smallest prime factor is
(Finch, 2001).
is the
-th primorial number.
* The OEIS considers 1 as a
-rough number for any
, although it doesn’t have a smallest prime factor (it is the empty product of primes).
. Numerator of density of integers with smallest prime factor prime (n). Numerator of density of prime (n)-rough numbers.
. Denominator of density of prime (n)-rough numbers.
. Denominator of density of integers with smallest prime factor prime (n).
-smooth numbers: largest prime factor of
(A006530)
. (Thus
-rough numbers.)
p |
p |
p |
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
(those are in arithmetic progression);2 k + 1, k ≥ 1 - 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 - ...
pn# |
n |
|
|
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
| ||
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
| ||
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
| ||
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
| ||
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
| ||
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
| ||
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
| ||
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
| ||
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
| ||
29 | {29, 31, 37, 41, 43, 47, ...} |
p |
p |
Density of prime (n)-rough numbers
The density of prime (n)-rough numbers is
{1, 1 −
|
{1,
|
where
{
|
{1 −
|
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
|
- {1, 1, 1, 4, 8, 16, 192, 3072, 55296, 110592, 442368, 13271040, 477757440, 19110297600, 802632499200, 1605264998400, 6421059993600, 12842119987200, 770527199232000, 50854795149312000, 3559835660451840000, ...}
|
- {1, 2, 3, 15, 35, 77, 1001, 17017, 323323, 676039, 2800733, 86822723, 3212440751, 131710070791, 5663533044013, 11573306655157, 47183480978717, 95993978542907, 5855632691117327, 392327390304860909, ...}
|
{2, 6, 15, 105, 385, 1001, 17017, 323323, 7436429, 19605131, 86822723, 3212440751, 131710070791, 5663533044013, 266186053068611, 613385252723321, 2783825377744303, 5855632691117327, 392327390304860909, ...}
Sequences
A063538 Not√ n − 1 |
n |
≥ √ n |
√ n |
- {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