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A239712
Primes of the form m = 2^i + 2^j - 1, where i > j >= 0.
14
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
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)
"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]
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