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

0,2

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

Links

Robert Israel, Table of n, a(n) for n = 0..10000

Peter Munn, Plot2 graph of a(n)/prime(n)

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A256967 A225249 A214048 * A088236 A194244 A014616

Adjacent sequences: A328656 A328657 A328658 * A328660 A328661 A328662

Keyword

nonn,base

Author

M. F. Hasler, Oct 25 2019

Status

approved