A345867 Total number of 0's in the binary expansions of the first n primes.

1, 1, 2, 2, 3, 4, 7, 9, 10, 11, 11, 14, 17, 19, 20, 22, 23, 24, 28, 31, 35, 37, 40, 43, 47, 50, 52, 54, 56, 59, 59, 64, 69, 73, 77, 80, 83, 87, 90, 93, 96, 99, 100, 105, 109, 112, 115, 116, 119, 122, 125, 126, 129, 130, 137, 142, 147, 151, 156, 161, 165, 170

Offset: 1

Author

Alois P. Heinz, Jun 26 2021

Keywords

• nonn
• base

Examples

			a(3) = 2: 2 = 10_2, 3 = 11_2, 5 = 101_2, so there are two 0's in the binary expansions of the first three primes.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

Partial sums of A035103.

Cf. A000040, A059305, A095375, A328659.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 0, a(n-1)
      +add(1-i, i=Bits[Split](ithprime(n))))
    end:
seq(a(n), n=1..100);

Mathematica

Accumulate[DigitCount[Prime[Range[100]], 2, 0]] (* Paolo Xausa, Feb 26 2024 *)

Python

from sympy import prime, primerange
from itertools import accumulate
def f(n): return (bin(n)[2:]).count('0')
def aupton(nn): return list(accumulate(map(f, primerange(2, prime(nn)+1))))
print(aupton(62)) # Michael S. Branicky, Jun 26 2021

Formula

a(n) = Sum_{i=1..n} A035103(i).

a(n) = a(n-1) for n in { A059305 }.

a(n) = A328659(n) - A095375(n).