A110299 a(n) = Sum_{i=0..n-1} 2^i*prime(n-i).

2, 7, 19, 45, 101, 215, 447, 913, 1849, 3727, 7485, 15007, 30055, 60153, 120353, 240759, 481577, 963215, 1926497, 3853065, 7706203, 15412485, 30825053, 61650195, 123300487, 246601075, 493202253, 986404613, 1972809335, 3945618783, 7891237693, 15782475517

Offset: 1

Author

Ryan Propper, Sep 07 2005

Keywords

• nonn
• easy

References

• Eric Angelini, "Array with primes." Pers. comm. on the SeqFan mailing list, Sep. 7 2005.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000040, A135483, A287353.

Cf. A125180, A000720, A007504

Programs

Magma

A110299:= func< n | (&+[2^(n-j)*NthPrime(j): j in [1..n]]) >;
[A110299(n): n in [1..40]]; // G. C. Greubel, Jan 03 2023

Maple

a:= proc(n) option remember;
      `if`(n=0, 0, ithprime(n)+2*a(n-1))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Dec 10 2016

Mathematica

Table[Sum[2^i * Prime[n-i], {i, 0, n-1}], {n, 1, 30}]

PARI

a(n) = fromdigits(primes(n),2); \\ Kevin Ryde, Jun 22 2022

Python

from sympy import prime
def A110299(n):
    c = 0
    for i in range(n):
        c = (c<<1)+prime(i+1)
    return c # Chai Wah Wu, Jan 04 2023

SageMath

@CachedFunction # a = A110299
def a(n): return 2 if (n==1) else 2*a(n-1) + nth_prime(n)
[a(n) for n in range(1,41)] # G. C. Greubel, Jan 03 2023

Formula

G.f: b(x)/(1-2*x), where b(x) is the g.f. of A000040. - Mario C. Enriquez, Dec 10 2016

a(n) = 2*a(n-1) + A000040(n) for n>0 with a(0)=0. - Alois P. Heinz, Dec 10 2016

From Ridouane Oudra, Jan 25 2024: (Start)

a(n) = Sum_{i=0..prime(n+1)-1} (2^(n-pi(i)) - 1), where prime(n) = A000040(n) and pi(n) = A000720(n).

a(n) = A125180(n+1) - A000040(n+1);

a(n) = Sum_{i=1..n} A125180(i);

a(n) = A007504(n) + Sum_{i=1..n-1} a(i). (End)