A239713 Primes of the form m = 3^i + 3^j - 1, where i > j >= 0.

3, 11, 29, 83, 89, 107, 251, 269, 809, 971, 2213, 2267, 6563, 6569, 6803, 8747, 19709, 19763, 20411, 59051, 65609, 177173, 183707, 531521, 538001, 590489, 1594331, 1594403, 1595051, 1596509, 4782971, 4782977, 4783697, 14348909, 14349149, 14526053, 14880347

Offset: 1

Author

Hieronymus Fischer, Mar 28 2014

Keywords

• nonn

Comments

The base-3 representation of a term 3^i + 3^j - 1 has base-3 digital sum = 1 + 2*j == 1 (mod 2).

In base-3 representation the first terms are 10, 102, 1002, 10002, 10022, 10222, 100022, 100222, 1002222, 1022222, 10000222, 10002222, 100000002, 100000022, 100022222, 102222222, 1000000222, 1000002222, 1000222222, 10000000002, 10022222222, 100000000222, 100022222222, ...

Examples

			a(1) = 3, since 3 = 3^1 + 3^0 - 1 is prime.
a(5) = 89, since 89 = 3^4 + 3^2 - 1 is prime.
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..131 [a(123) corrected by _Georg Fischer_, Dec 22 2024]

Crossrefs

Cf. A018900, A239709, A239712 (base 2), A239714 (base 4), A239715 (base 5), A239716 (base 6), A239717 (base 7), A239718 (base 8), A239719 (base 9), A239720 (base 10).

Programs

Maple

select(isprime, [seq(seq(3^i+3^j-1, j=0..i-1), i=1..25)])[];  # Alois P. Heinz, Dec 22 2024

Mathematica

Select[Flatten[Table[3^i + 3^j - 1, {i, 1, 25}, {j, 0, i - 1}]], PrimeQ] (* Jean-François Alcover, Mar 17 2025, after Alois P. Heinz *)

Smalltalk

A239713
"Answers the n-th term of A239713.
  Usage: n A239713
  Answer: a(n)"
  | a b i j k p q terms |
  terms := OrderedCollection new.
  k := 0.
  b := 3.
  p := b.
  i := 1.
  [k < self] whileTrue:
         [j := 0.
         q := 1.
         [j < i and: [k < self]] whileTrue:
                   [a := p + q - 1.
                   a isPrime
                        ifTrue:
                            [k := k + 1.
                            terms add: a].
                   q := b * q.
                   j := j + 1].
         i := i + 1.
         p := b * p].
     ^terms at: self
[by Hieronymus Fischer, Apr 14 2014]
--------------------

Smalltalk

A239713
"Version 2: Answers the n-th term of A239713.
  Uses distinctPowersOf: b from A018900
  Usage: n A239713
  Answer: a(n)”
  | a k n terms |
  terms := OrderedCollection new.
  n := 1.
  k := 0.
  [k < self] whileTrue:
         [(a:= (n distinctPowersOf: 3) - 1)
              isPrime ifTrue:    [k := k + 1.
                                 terms add: a].
              n := n + 1].
  ^terms at: self
[by Hieronymus Fischer, Apr 22 2014]
-----------

Smalltalk

A239713
  "Version 3: Answer an array of the first n terms of A239713.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
  Usage: n A239713
  Answer: #(3 11 29 ... ) [a(1) ... a(n)]”
  ^self primesWhichAreDistinctPowersOf: 3 withOffset: -1
[by Hieronymus Fischer, Apr 22 2014]