A239712 Primes of the form m = 2^i + 2^j - 1, where i > j >= 0.

2, 5, 11, 17, 19, 23, 47, 67, 71, 79, 131, 191, 257, 263, 271, 383, 1031, 1039, 1087, 1151, 1279, 2063, 2111, 4099, 4111, 4127, 4159, 5119, 6143, 8447, 16447, 20479, 32771, 32783, 32831, 33023, 33791, 65537, 65539, 65543, 65551, 65599, 66047, 73727, 81919, 262147, 262151, 262271, 262399, 263167

Offset

1,1

Comments

Numbers m such that b = 2 is the only base such that the base-b digital sum of m + 1 is equal to b.

Example: 5 + 1 = 110_2 which implies ds_2(5 + 1) = 2 = b, where ds_b = digital sum in base-b. However, ds_3(6) = 2 <> 3, ds_4(6) = 3 <> 4, ds_5(6) = 2 <> 5, ds_6(6) = 1 <> 6. For all other bases > 6 we have ds_b(6) = 6 <> b. It follows that b = 2 is the only such base.

The base-2 representation of a term 2^i + 2^j - 1 has a base-2 digital sum of 1 + j.

In base-2 representation the first terms are 10, 101, 1011, 10001, 10011, 10111, 101111, 1000011, 1000111, 1001111, 10000011, 10111111, 100000001, 100000111, 100001111, 101111111, 10000000111, 10000001111, 10000111111, 10001111111, ...

Numbers m = 2^i + 2^j - 1 with odd i and j are not terms. Example: 10239 = 2^13 + 2^11 - 1 is not a prime.

Links

Hieronymus Fischer, Table of n, a(n) for n = 1..250

Formula

a(n) = A239708(n) - 1.

a(n+1) = min(A018900(k) > a(n)| A018900(k) - 1 is prime, k >= 1) - 1.

Example

a(1) = 2, since 2 = 2^1 + 2^0 - 1 is prime.
a(5) = 19, since 19 = 2^4 + 2^2 - 1 is prime.

Mathematica

Select[Union[Total/@(2^#&/@Subsets[Range[0, 20], {2}])-1], PrimeQ] (* Harvey P. Dale, Aug 08 2014 *)

Prog

(Smalltalk)
A239712
"Answers the n-th term of A239712.
  Usage: n A239712
  Answer: a(n)"
  | a b i k m p q terms |
  terms := OrderedCollection new.
  b := 2.
  p := 1.
  k := 0.
  m := 0.
  [k < self] whileTrue:
         [m := m + 1.
         p := b * p.
         q := 1.
         i := 0.
         [i < m and: [k < self]] whileTrue:
                   [i := i + 1.
                   a := p + q - 1.
                   a isPrime
                        ifTrue:
                            [k := k + 1.
                            terms add: a].
                   q := b * q]].
  ^terms at: self
[by Hieronymus Fischer, Apr 22 2014]
-----------
(Smalltalk)
floorPrimesWhichAreDistinctPowersOf: b withOffset: d
  "Answers an array which holds the primes < n that obey b^i + b^j + d, i>j>=0,
  where n is the receiver. b > 1 (here: b = 2, d = -1).
  Uses floorDistinctPowersOf: from A018900
  Usage:
  n floorPrimesWhichAreDistinctPowersOf: b withOffset: d
  Answer: #(2 5 11 17 19 23 ...) [terms < n]"
  ^((self - d floorDistinctPowersOf: b)
  collect: [:i | i + d]) select: [:i | i isPrime]
[by Hieronymus Fischer, Apr 22 2014]
------------
(Smalltalk)
primesWhichAreDistinctPowersOf: b withOffset: d
  "Answers an array which holds the n primes of the form b^i + b^j + d, i>j>=0, where n is the receiver.
  Direct calculation by scanning b^i + b^j + d in increasing order and selecting terms which are prime.
  b > 1; this sequence: b = 2, d = 1.
  Usage:
  n primesWhichAreDistinctPowersOf: b withOffset: d
  Answer: #(2 5 11 17 19 23 ...) [a(1) ... a(n)]"
  | a k p q terms n |
  terms := OrderedCollection new.
  n := self.
  k := 0.
  p := b.
  [k < n] whileTrue:
         [q := 1.
         [q < p and: [k < n]] whileTrue:
                   [a := p + q + d.
                   a isPrime
                        ifTrue:
                            [k := k + 1.
                            terms add: a].
                   q := b * q].
         p := b * p].
  ^terms asArray
[by Hieronymus Fischer, Apr 22 2014]

Crossrefs

Cf. A007953, A018900, A081091, A008864, A187813.

Cf. A239703, A239708, A239709, A239713 - A239720.

Sequence in context: A070957 A166744 A080165 * A224363 A307508 A063535

Adjacent sequences: A239709 A239710 A239711 * A239713 A239714 A239715

Keyword

nonn

Author

Hieronymus Fischer, Mar 28 2014 and Apr 22 2014

Extensions

Examples moved from Maple field to Examples field by Harvey P. Dale, Aug 08 2014

Status

approved