A328659 -id:A328659 - cp's OEIS frontend

A154703 Concatenation of the first n primes written in base 2.

10, 1011, 1011101, 1011101111, 10111011111011, 101110111110111101, 10111011111011110110001, 1011101111101111011000110011, 101110111110111101100011001110111, 10111011111011110110001100111011111101, 1011101111101111011000110011101111110111111

Offset: 1

Jonathan Vos Post, Jan 14 2009

Number of bits in a(n) = Sum_{i=1..n} A035100(i). See A328659(n).

			a(2) = 1011 = 10 Concat 11 = (2 base 2) Concat (3 base 2).

Seiichi Manyama, Table of n, a(n) for n = 1..122

Cf. A000040, A004676, A007088, A035100, A328659.

A154703 := proc(n) option remember: local d: if(n=1)then return 10: fi: d:=convert(ithprime(n),base,2): return parse(cat(convert(procname(n-1),string), convert(op(convert(d,base,10,10^nops(d))),string))): end: seq(A154703(n),n=1..10); # Nathaniel Johnston, May 27 2011
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 0,
      parse(cat(a(n-1), convert(ithprime(n), binary))))
    end:
seq(a(n), n=1..12);  # Alois P. Heinz, Dec 16 2024

With[{p = IntegerDigits[Prime[Range[15]], 2]}, Array[FromDigits[Flatten[p[[;;#]]]] &, Length[p]]] (* Paolo Xausa, Feb 26 2024 *)

a(n) = fromdigits(concat(apply(binary, primes(n)))); \\ Michel Marcus, Jul 17 2023

a(n) = CONCATENATE[i=1..n] A004676(i) = CONCATENATE[i=1..n] A007088(A000040(i)).

Corrected terms a(6) and beyond from Seiichi Manyama, Jul 17 2023

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

Original entry on oeis.org

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

Alois P. Heinz, Jun 26 2021

			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.

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

Partial sums of A035103.

Cf. A000040, A059305, A095375, A328659.

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);

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

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

a(n) = Sum_{i=1..n} A035103(i).
a(n) = a(n-1) for n in { A059305 }.
a(n) = A328659(n) - A095375(n).

cp's OEIS Frontend

A154703 Concatenation of the first n primes written in base 2.

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