C#, 1923
It probably wont be the shortest program but i found the challenge interesting, so here is my solution.
Running all 4 with 35 Numbers (15 35) takes about 5 seconds.
You can test it here, but note that if you want OEIS4 the amount of digits you want needs to be small or netfiddle runs out of memory.
Golfed
using System;using System.Collections;using System.Collections.Generic;using System.Linq;class p{public static void Main(string[] args){int b=0;IEnumerable<int>a=null;foreach(char c in Convert.ToString(int.Parse(args[0]),2).Reverse()){++b;if(c=='0')continue;switch(b){case 1: a=d(a,e());break;case 2: a=d(a,f());break;case 3: a=d(a,g());break;case 4: a=d(a,h(),true);break;}}if(a==null)return;bool j=true;foreach(int i in a.Take(int.Parse(args[1]))){if(j)j=false;else Console.Write(",");Console.Write(i);}}static IEnumerable<int>d(IEnumerable<int>k,IEnumerable<int>l,bool m=false){if(k==null)foreach(int n in l)yield return n;int o=0;int p=1;foreach(int i in k){Dictionary<int,HashSet<int>>q=m ? new Dictionary<int,HashSet<int>>(): null;int s=0;foreach(int n in l){if(!m){if(i<n)break;}else{if(!q.ContainsKey(o))q.Add(o,new HashSet<int>());q[o].Add(n);if(q.Count==1){int r=q[o].OrderBy(gi =>gi).Take(2).Sum();if(i<r)break;}else{int r=q[o].Concat(q[o-1]).OrderBy(gi =>gi).Take(2).Sum();if(i<r)break;}if(++s==p){o++;p=(int)Math.Pow(2,o);}}if(i==n){yield return i;break;}}}}static IEnumerable<int>e(){int t=0;for(int i=0;i<int.MaxValue;i++)foreach(char c in Convert.ToString(i,2)){if(c=='0')yield return t;t++;}}static IEnumerable<int>f(){int t=1;int u=0;bool v=true;using(IEnumerator<int>w=Enumerable.Range(0,int.MaxValue).GetEnumerator()){while(w.MoveNext()){if(v){if(u==0)u=t+1;yield return w.Current;if(--t==0)v=false;}else{if(t==0)t=u+1;if(--u==0)v=true;}}}}static IEnumerable<int>g(){for(int i=0;i<int.MaxValue;i++){string s=Convert.ToString(i,2);if(x(s.Count(c =>c=='0'))&& x(s.Count(c =>c=='1')))yield return i;}}static bool x(int y){return(y != 0)&&((y &(y-1))==0);}static IEnumerable<int>h(){return Enumerable.Range(1,int.MaxValue).Select(z);}static Dictionary<int,int>_=new Dictionary<int,int>();static int z(int n){int a;if(!_.TryGetValue(n,out a)){if(n<3)a=1;else a=z(n-z(n-1))+z(n-z(n-2));_.Add(n,a);}return a;}}
Readable
using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
class Programm
{
public static void Main(string[] args)
{
int index = 0;
IEnumerable<int> intersection = null;
foreach (char c in Convert.ToString(int.Parse(args[0]), 2).Reverse())
{
++index;
if (c == '0')
continue;
switch (index)
{
case 1: intersection = _join(intersection, OEIS1()); break;
case 2: intersection = _join(intersection, OEIS2()); break;
case 3: intersection = _join(intersection, OEIS3()); break;
case 4: intersection = _join(intersection, OEIS4(), true); break;
default: throw new ArgumentException();
}
}
if (intersection == null)
return;
bool first = true;
foreach (int i in intersection.Take(int.Parse(args[1])))
{
if (first) first = false;
else Console.Write(",");
Console.Write(i);
}
Console.ReadKey();
}
private static IEnumerable<int> _join(IEnumerable<int> intersection, IEnumerable<int> newSequence, bool hof = false)
{
if (intersection == null)
foreach (int n in newSequence) yield return n;
int generation = 0;
int generationMax = 1;
foreach (int i in intersection)
{
Dictionary<int, HashSet<int>> generationCache = hof ? new Dictionary<int, HashSet<int>>() : null;
int count = 0;
foreach (int n in newSequence)
{
if (!hof)
{
if (i < n)
break;
}
else
{
if (!generationCache.ContainsKey(generation))
generationCache.Add(generation, new HashSet<int>());
generationCache[generation].Add(n);
if (generationCache.Count == 1)
{
int lowerBound = generationCache[generation].OrderBy(gi => gi).Take(2).Sum();
if (i < lowerBound)
break;
}
else
{
int lowerBound = generationCache[generation].Concat(generationCache[generation - 1]).OrderBy(gi => gi).Take(2).Sum();
if (i < lowerBound)
break;
}
if (++count == generationMax)
{
generation++;
generationMax = (int)Math.Pow(2, generation);
}
}
if (i == n)
{
yield return i;
break;
}
}
}
}
static IEnumerable<int> OEIS1()
{
int position = 0;
for (int i = 0; i < int.MaxValue; i++)
foreach (char c in Convert.ToString(i, 2))
{
if (c == '0')
yield return position;
position++;
}
}
static IEnumerable<int> OEIS2()
{
int take = 1;
int skip = 0;
bool doTake = true;
using (IEnumerator<int> enumerator = Enumerable.Range(0, int.MaxValue).GetEnumerator())
{
while (enumerator.MoveNext())
{
if (doTake)
{
if (skip == 0)
skip = take + 1;
yield return enumerator.Current;
if (--take == 0)
doTake = false;
}
else
{
if (take == 0)
take = skip + 1;
if (--skip == 0)
doTake = true;
}
}
}
}
static IEnumerable<int> OEIS3()
{
for (int i = 0; i < int.MaxValue; i++)
{
string s = Convert.ToString(i, 2);
if (_isPowerOfTwo(s.Count(c => c == '0')) && _isPowerOfTwo(s.Count(c => c == '1')))
yield return i;
}
}
static bool _isPowerOfTwo(int number)
{
return (number != 0) && ((number & (number - 1)) == 0);
}
static IEnumerable<int> OEIS4()
{
return Enumerable.Range(1, int.MaxValue).Select(HofstadterQ);
}
static Dictionary<int, int> _hofstadterQCache = new Dictionary<int, int>();
static int HofstadterQ(int n)
{
int result;
if (!_hofstadterQCache.TryGetValue(n, out result))
{
if (n < 3)
result = 1;
else
result = HofstadterQ(n - HofstadterQ(n - 1)) + HofstadterQ(n - HofstadterQ(n - 2));
_hofstadterQCache.Add(n, result);
}
return result;
}
}
Explanation
This makes use of lazy evaluation bigtime which makes it prett fast i bellieve. Also i was lazy doing any "bitlogic" by using the frameworks Convert.ToString(number, 2) method. This turns any number into its binray representation as a string.
I had to write my own method to intersect the seuqences as the Linq-Method intersect computes the intersection of the full sequence, and that was literally impossible.
12 5example up to the same index, then10does indeed come before9in the intersection... like, how would you, while going through the sequences, decide whether to skip the9in #3 as a possible intersection? Like if #3 had7in it then you would be required to skip over it since that doesn't appear in #4 \$\endgroup\$x? \$\endgroup\$