# Smallest number k such that k+2^m is composite for all m less than k

Generate the sequence of numbers a(k), where each k is the smallest positive integer such that k + 2m is composite for every positive integer m less than k.

Smallest number k such that k+2^m is composite for all m less than k
You are encouraged to solve this task according to the task description, using any language you may know.

For example

Suppose k == 7; test m == 1 through m == 6. If any are prime, the test fails.

Is 7 + 21 (9) prime? False

Is 7 + 22 (11) prime? True

So 7 is not an element of this sequence.

It is only necessary to test odd natural numbers k. An even number, plus any positive integer power of 2 is always composite.

Find and display, here on this page, the first 5 elements of this sequence.

## Go

Translation of: Wren

Takes around 2.2 seconds though faster than using Go's native big.Int type which takes 6.2 seconds.

```package main

import (
"fmt"
big "github.com/ncw/gmp"
)

// returns true if k is a sequence member, false otherwise
func a(k int64) bool {
if k == 1 {
return false
}
bk := big.NewInt(k)
for m := uint(1); m < uint(k); m++ {
n := big.NewInt(1)
n.Lsh(n, m)
if n.ProbablyPrime(15) {
return false
}
}
return true
}

func main() {
count := 0
k := int64(1)
for count < 5 {
if a(k) {
fmt.Printf("%d ", k)
count++
}
k += 2
}
fmt.Println()
}
```
Output:
```773 2131 2491 4471 5101
```

## Julia

```using Lazy
using Primes

a(k) = all(m -> !isprime(k + big"2"^m), 1:k-1)

A033939 = @>> Lazy.range(2) filter(isodd) filter(a)

println(take(5, A033939))   # List: (773 2131 2491 4471 5101)
```

## Mathematica/Wolfram Language

Since the code is reasonably performant I found the first 8 of this sequence:

```ClearAll[ValidK]
ValidK[1] := False
ValidK[k_] := If[EvenQ[k],
False,
AllTrue[Range[k - 1], CompositeQ[k + 2^#] &]
]
list = {};
Do[
If[ValidK[k],
AppendTo[list, k];
If[Length[list] >= 8, Break[]]
]
,
{k, 1, \[Infinity]}
]
list
```
Output:
`{773, 2131, 2491, 4471, 5101, 7013, 8543, 10711}`

## Perl

Library: ntheory
```use strict;
use warnings;
use bigint;
use ntheory 'is_prime';

my \$cnt;
LOOP: for my \$k (2..1e10) {
next unless 1 == \$k % 2;
for my \$m (1..\$k-1) {
next LOOP if is_prime \$k + (1<<\$m)
}
print "\$k ";
last if ++\$cnt == 5;
}
```
Output:
`773 2131 2491 4471 5101`

## Phix

```with javascript_semantics
atom t0 = time()
include mpfr.e

mpz z = mpz_init()
function a(integer k)
if k=1 then return false end if
for m=1 to k-1 do
mpz_ui_pow_ui(z,2,m)
if mpz_prime(z) then return false end if
end for
return true
end function

integer k = 1, count = 0
while count<5 do
if a(k) then
printf(1,"%d ",k)
count += 1
end if
k += 2
end while
printf(1,"\n")
?elapsed(time()-t0)
```
Output:

Rather slow, even worse under pwa/p2js - about 90s...

```773 2131 2491 4471 5101
"22.7s"
```

## Raku

```put (1..∞).hyper(:250batch).map(* × 2 + 1).grep( -> \$k { !(1 ..^ \$k).first: (\$k + 1 +< *).is-prime } )[^5]
```
Output:
`773 2131 2491 4471 5101`

## Wren

Library: Wren-gmp

An embedded version as, judging by the size of numbers involved, Wren-CLI (using BigInt) will be too slow for this.

Brute force approach - takes a smidge under 2 seconds.

```import "./gmp" for Mpz

// returns true if k is a sequence member, false otherwise
var a = Fn.new { |k|
if (k == 1) return false
for (m in 1...k) {
if (n.probPrime(15) > 0) return false
}
return true
}

var count = 0
var k = 1
while (count < 5) {
if (a.call(k)) {
System.write("%(k) ")
count = count + 1
}
k = k + 2
}
System.print()
```
Output:
```773 2131 2491 4471 5101
```