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A370069 Lexicographically earliest sequence of distinct integers such that the concatenated binary expansions of the terms is A010051.

%I A370069 #18 Mar 19 2024 16:38:10
%S A370069 0,1,2,40,162,8,32,160,34,544,130,520,2568,8320,552,663552,2178,512,
%T A370069 10272,34848,2560,665600,2048,35360,163872,2080,10274,8396800,9052160,
%U A370069 33280,2592,128,33288,133128,131584,10242,33312,2056,165888,526464,2230272,655360,2129952,8352,32800,534560,141312,2050,139394,32776
%N A370069 Lexicographically earliest sequence of distinct integers such that the concatenated binary expansions of the terms is A010051.
%C A370069 If we take the binary expansion of each term and concatenate these bits to a sequence, we get the sequence of the characteristic function of primes (A010051).
%C A370069 For n > 2 every term is an even Fibbinary number (A022340).
%H A370069 Michael S. Branicky, <a href="/A370069/b370069.txt">Table of n, a(n) for n = 1..10000</a>
%e A370069 terms   0,   1,    2,         40,             162,            8,          32
%e A370069 binary {0}, {1}, {1,0}, {1,0,1,0,0,0}, {1,0,1,0,0,0,1,0}, {1,0,0,0}, {1,0,0,0,0,0}
%e A370069 A010051 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0
%t A370069 n=49; lst={0};p=2;c=Boole[PrimeQ@Range[n^2]]; Do[k=1;While[MemberQ[lst,ne=FromDigits[c[[p;;(pn=NextPrime[p,k])-1]],2]],k++]; AppendTo[lst,ne];p=pn,{i,n}];lst
%o A370069 (Python)
%o A370069 from sympy import nextprime
%o A370069 from itertools import islice
%o A370069 def agen(): # generator of terms
%o A370069     yield 0
%o A370069     p, nextp, aset = 2, 3, {0}
%o A370069     while True:
%o A370069         an = 0
%o A370069         while an in aset:
%o A370069             an = (an<<(nextp-p)) + (1<<(nextp-p-1))
%o A370069             p, nextp = nextp, nextprime(nextp)
%o A370069         yield an
%o A370069         aset.add(an)
%o A370069 print(list(islice(agen(), 50))) # _Michael S. Branicky_, Feb 08 2024
%Y A370069 Cf A010051, A022340, A003714, A030190, A139102.
%K A370069 nonn,base
%O A370069 1,3
%A A370069 _Giorgos Kalogeropoulos_, Feb 08 2024