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

0, 1, 2, 40, 162, 8, 32, 160, 34, 544, 130, 520, 2568, 8320, 552, 663552, 2178, 512, 10272, 34848, 2560, 665600, 2048, 35360, 163872, 2080, 10274, 8396800, 9052160, 33280, 2592, 128, 33288, 133128, 131584, 10242, 33312, 2056, 165888, 526464, 2230272, 655360, 2129952, 8352, 32800, 534560, 141312, 2050, 139394, 32776

Offset

1,3

Comments

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).

For n > 2 every term is an even Fibbinary number (A022340).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

terms   0,   1,    2,         40,             162,            8,          32
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}
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

Mathematica

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

Prog

(Python)
from sympy import nextprime
from itertools import islice
def agen(): # generator of terms
    yield 0
    p, nextp, aset = 2, 3, {0}
    while True:
        an = 0
        while an in aset:
            an = (an<<(nextp-p)) + (1<<(nextp-p-1))
            p, nextp = nextp, nextprime(nextp)
        yield an
        aset.add(an)
print(list(islice(agen(), 50))) # Michael S. Branicky, Feb 08 2024

Crossrefs

Cf A010051, A022340, A003714, A030190, A139102.

Sequence in context: A275547 A269544 A281761 * A359859 A012477 A003760

Adjacent sequences: A370066 A370067 A370068 * A370070 A370071 A370072

Keyword

nonn,base

Author

Giorgos Kalogeropoulos, Feb 08 2024

Status

approved