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# P-rough numbers

(Redirected from 31-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):

where
 pn#
is the
 n
-th primorial number.

## Table of sequences

NOTE: 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).

 p
-rough numbers sequences

 p
 a (n), n   ≥   1.
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, ...} A??????
31 {31, 37, 41, 43, 47, ...} A??????

## 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, ...}