A239714 -id:A239714 - cp's OEIS frontend

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

Hieronymus Fischer, Mar 28 2014

nonn

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, ...

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

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

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).

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

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 *)

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]
--------------------

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]
-----------

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]

A239718 Primes of the form m = 8^i + 8^j - 1, where i > j >= 0.

Original entry on oeis.org

71, 4159, 32831, 262151, 266239, 294911, 2101247, 18874367, 134479871, 1073741831, 68721573887, 549755813951, 4398046515199, 4398046543871, 4398046773247, 4398063288319, 281474976711167, 281474976743423, 281474978807807, 281474993487871, 282024732524543

Offset: 1

Hieronymus Fischer, Apr 14 2014

nonn

The base-8 representation of a term 8^i + 8^j - 1 has base-8 digital sum = 1 + 7*j == 1 (mod 7).
In base-8 representation the first terms are 107, 10077, 100077, 1000007, 1007777, 1077777, 10007777, 107777777, 1000777777, 10000000007, 1000007777777, 10000000000077, 100000000007777, ...
Numbers m that satisfy m = 8^i + 8^j - 1 with odd i and j are not terms. Example: 33279 = 8^5 + 8^3 - 1 = 3*11093.

			a(1) = 71, since 71 = 8^2 + 8^1 - 1 is prime.
a(2) = 4159, since 4159 = 8^4 + 8^2 - 1 is prime.

Georg Fischer, Table of n, a(n) for n = 1..40 [First 35 terms from Hieronymus Fischer]

Cf. A018900, A239709, A239712 (base 2), A239713 (base 3), A239714 (base 4), A239715 (base 5), A239716 (base 6), A239717 (base 7), A239719 (base 9), A239720 (base 10).

select(isprime, [seq(seq(8^i+8^j-1, j=0..i-1), i=1..25)])[];  # Alois P. Heinz, Dec 22 2024

A239718
  "Answers an array of the first n terms of A239718.
  Uses method primesWhichAreDistinctPowersOf: b withOffset: d from A239712.
Usage: n A239718
Answer: #(71 4159 ... ) [a(1) ... a(n)]"
  ^self primesWhichAreDistinctPowersOf: 8 withOffset: -1

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

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A239718 Primes of the form m = 8^i + 8^j - 1, where i > j >= 0.

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Programs

Maple

Smalltalk