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

0,2

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

Links

Table of n, a(n) for n=0..11.

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 *)

Prog

(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

Crossrefs

Cf. A095375.

Sequence in context: A297633 A152279 A015575 * A228421 A152109 A222064

Adjacent sequences: A214341 A214342 A214343 * A214345 A214346 A214347

Keyword

nonn,more,base

Author

Tjandra Satria Gunawan, Jul 13 2012

Extensions

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

Status

approved