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Difference between revisions of "Williams prime"

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(OEISs)
 
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! b !! MM<br>{{Kbn|(b-1)|b|n}} !! MP<br>{{Kbn|+|(b-1)|b|n}} !! PM<br>{{Kbn|(b+1)|b|n}} !! PP<br>{{Kbn|+|(b+1)|b|n}}
 
! b !! MM<br>{{Kbn|(b-1)|b|n}} !! MP<br>{{Kbn|+|(b-1)|b|n}} !! PM<br>{{Kbn|(b+1)|b|n}} !! PP<br>{{Kbn|+|(b+1)|b|n}}
 
|-
 
|-
| 2 || {{OEIS|A000043}} || none || {{OEIS|A002235}} || {{OEIS|A002253}}
+
| 2 || {{OEIS|A000043}} -> [[Williams prime MM 2|page]] || || {{OEIS|A002235}} -> [[Williams prime PM 2|page]] || {{OEIS|A002253}} -> [[Williams prime PP 2|page]]
 
|-
 
|-
| 3 || {{OEIS|A003307}} || {{OEIS|A003306}} || {{OEIS|A005540}} || {{OEIS|A005537}}
+
| 3 || {{OEIS|A003307}} -> [[Williams prime MM 3|page]] || {{OEIS|A003306}} -> [[Williams prime MP 3|page]] || {{OEIS|A005540}} -> [[Williams prime PM 3|page]] || {{OEIS|A005537}} -> [[Williams prime PP 3|page]]
 
|-
 
|-
| 4 || {{OEIS|A272057}} || none || none|| no primes
+
| 4 || {{OEIS|A272057}} -> [[Williams prime MM 4|page]] || {{OEIS|A326655}} -> [[Williams prime MP 4|page]] || ||
 
|-
 
|-
| 5 || {{OEIS|A046865}} || {{OEIS|A204322}} || {{OEIS|A257790}} || {{OEIS|A143279}}
+
| 5 || {{OEIS|A046865}} -> [[Williams prime MM 5|page]] || {{OEIS|A204322}} -> [[Williams prime MP 5|page]] || {{OEIS|A257790}} -> [[Williams prime PM 5|page]] || {{OEIS|A143279}} -> [[Williams prime PP 5|page]]
 
|-
 
|-
| 6 || {{OEIS|A079906}} || {{OEIS|A247260}} || none || none
+
| 6 || {{OEIS|A079906}} -> [[Williams prime MM 6|page]] || {{OEIS|A247260}} -> [[Williams prime MP 6|page]] || ||
 
|-
 
|-
| 7 || {{OEIS|A046866}} || {{OEIS|A245241}} || none || no primes
+
| 7 || {{OEIS|A046866}} -> [[Williams prime MM 7|page]] || {{OEIS|A245241}} -> [[Williams prime MP 7|page]] || ||
 
|-
 
|-
| 8 || {{OEIS|A268061}} || {{OEIS|A269544}} || none || none
+
| 8 || {{OEIS|A268061}} -> [[Williams prime MM 8|page]] || {{OEIS|A269544}} -> [[Williams prime MP 8|page]] || ||
 
|-
 
|-
| 9 || {{OEIS|A268356}} || {{OEIS|A056799}} || none || none
+
| 9 || {{OEIS|A268356}} -> [[Williams prime MM 9|page]] || {{OEIS|A056799}} -> [[Williams prime MP 9|page]] || ||
 
|-
 
|-
| 10 || {{OEIS|A056725}} || {{OEIS|A056797}} || {{OEIS|A111391}} || no primes
+
| 10 || {{OEIS|A056725}} -> [[Williams prime MM 10|page]] || {{OEIS|A056797}} -> [[Williams prime MP 10|page]] || {{OEIS|A111391}} -> [[Williams prime PM 10|page]] ||
 +
|-
 +
| 11 || {{OEIS|A046867}} -> [[Williams prime MM 11|page]] || {{OEIS|A057462}} -> [[Williams prime MP 11|page]] || ||
 +
|-
 +
| 12 || {{OEIS|A079907}} -> [[Williams prime MM 12|page]] || {{OEIS|A251259}} -> [[Williams prime MP 12|page]] || ||
 +
|-
 +
| 13 || {{OEIS|A297348}} -> [[Williams prime MM 13|page]] || || ||
 +
|-
 +
| 14 || {{OEIS|A273523}} -> [[Williams prime MM 14|page]] || || ||
 
|}
 
|}
  

Latest revision as of 07:30, 5 August 2019

Definition

A Williams number is a natural number of the form (b-1)bn-1 for integers b ≥ 2 and n ≥ 1.

A Williams prime is a Williams number which is prime.

Generalization

Varying both signs, there're four different types of numbers similiar to Williams numbers.

Lists of primes for bases b and n-values can be found here:

Type Category[1] Table [2] Smallest [3] Remaining[4]
MM: (b-1)bn-1 here here
169 bases
here[5]
40 unknown
here
MP: (b-1)bn+1 here here
153 bases
here
21 unknown
here
PM: (b+1)bn-1 here here
87 bases
here
1 unknown
here
PP: (b+1)bn+1 here here
82 bases
here [6]
4 unknown
here
  1. Containing all related pages for the type.
  2. The table contains only bases which are included as a separate page.
  3. The list of smallest primes of any base is an ASCII file for 2 ≤ b ≤ 1024. For unknown values only the base is given.
  4. All data not yet available as separate page.
  5. The list contains values for 2 ≤ b ≤ 2049.
  6. Values for bases b ≡ 1 mod 3 are always divisible by 3, so not listed here.

Available Online Sequences

Here are listed the available sequences in the On-Line Encyclopedia of Integer Sequences.

External links

Number classes
General numbers
Special numbers
Prime numbers