A328659 Partial sums of A035100: number of binary digits of the primes.

0, 2, 4, 7, 10, 14, 18, 23, 28, 33, 38, 43, 49, 55, 61, 67, 73, 79, 85, 92, 99, 106, 113, 120, 127, 134, 141, 148, 155, 162, 169, 176, 184, 192, 200, 208, 216, 224, 232, 240, 248, 256, 264, 272, 280, 288, 296, 304, 312, 320, 328, 336, 344, 352, 360, 369, 378, 387, 396, 405, 414, 423, 432

Offset: 0

Author

M. F. Hasler, Oct 25 2019

Keywords

• nonn
• base

Comments

Useful to express the binary Copeland-Erdős constant, cf. formula.

Plotting a(n) against prime(n) might be a good tool for introducing students of mathematics, particularly those who are familiar with the use of binary representation, to the way the density of prime numbers decreases with increasing size. In essence, the graph of a(n) against prime(n) is approximately linear, and this becomes more obvious if we plot a(n)/prime(n): see the relevant plot in the links. - Peter Munn, Mar 03 2024

Examples

			Primes written in binary (A004676) read: 10, 11, 101, 111, 1011, 1101, 10001, ...
The length of the concatenation of the first n = 0, 1, 2, 3, .... terms is
  0, 2, 4, 7, 10, 14, 18, 23, ...: this sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000
• Peter Munn, Plot2 graph of a(n)/prime(n)

Crossrefs

Cf. A004676 (primes in binary), A035100 (their number of digits), A066747 & A191232: decimals and bits of the binary Copeland-Erdős constant.

Cf. A095375, A345867.

Programs

Maple

a := n -> add(ilog2(ithprime(k)), k=1..n) + n:
seq(a(n), n=0..62); # Peter Luschny, Oct 26 2019

Mathematica

Join[{0}, Accumulate[BitLength[Prime[Range[100]]]]] (* Paolo Xausa, Mar 20 2024 *)

PARI

s=0; A328659=vector(50,n,s+=logint(prime(n),2)+1)

Python

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

Formula

a(n) = n + Sum_{k=1..n} floor(log_2(prime(k))).

A066747 = Sum_{n >= 1} prime(n)/2^a(n), the binary Copeland-Erdős constant.

a(n) = a(n-1) + A035100(n), n >= 1.

a(n) = A095375(n) + A345867(n) for n >= 1. - Alois P. Heinz, Jun 28 2021