A214344 Number of 1's in the first 10^n binary digits in the stream of prime numbers in base 2.

1, 8, 69, 593, 5723, 56090, 541794, 5369528, 53803123, 527428642, 5249946808, 52800311682

Offset: 0

Author

Tjandra Satria Gunawan, Jul 13 2012

Keywords

• nonn
• more
• base

Comments

Consider the stream (concatenation) of binary digits of primes in the MSB-first order featured in A191232. a(n) is the total count of 1's in the first 10^n of zeros and ones in this stream.

The complementary count of 0's is 10^n - a(n) = 0, 2, 31, 407, 4277, 43910, 458206, ... - R. J. Mathar, Jul 16 2012

Crossrefs

Cf. A095375.

Programs

Maple

A214344 := proc()
    local stre,len,ct,p ;
    stre := [] ;
    len := 2 ;
    ct := 1 ;
    p := 2 ;
    while true do
        if nops(stre) = 0 then
            p := nextprime(p) ;
            stre := convert(p,base,2) ;
        end if;
        if op(-1,stre) = 1 then
            ct := ct+ 1;
        end if;
        stre := subsop(-1=NULL,stre) ;
        len := len+1 ;
        if ilog10(len-1) <> ilog10(len) then
            print(ct) ;
        end if;
    end do:
end proc: # R. J. Mathar, Jul 14 2012

Mathematica

pow = 1; sum1 = 0; sum2 = 0; p = 2;seq={}; k = 0; Do[d = IntegerDigits[p, 2]; sum1 += Count[d, 1]; sum2 += Length[d]; k++; If[sum2 >= pow, del = sum2 - pow; term = sum1 - Count[d[[-del ;; -1]], 1];   AppendTo[seq, term]; pow *= 10]; p = NextPrime[p], {10^4}]; seq (* Amiram Eldar, May 10 2019 *)

Python

from sympy import nextprime
from itertools import islice
def bgen(p=2):
    while True: yield from (int(b) for b in bin(p)[2:]); p = nextprime(p)
def a(n): return sum(islice(bgen(), 10**n))
print([a(n) for n in range(7)]) # Michael S. Branicky, Jul 03 2022

Extensions

a(9)-a(11) from Amiram Eldar, May 10 2019