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A214344
Number of 1's in the first 10^n binary digits in the stream of prime numbers in base 2.
1
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
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
KEYWORD
nonn,more,base
AUTHOR
EXTENSIONS
a(9)-a(11) from Amiram Eldar, May 10 2019
STATUS
approved