A001751 Primes together with primes multiplied by 2.

2, 3, 4, 5, 6, 7, 10, 11, 13, 14, 17, 19, 22, 23, 26, 29, 31, 34, 37, 38, 41, 43, 46, 47, 53, 58, 59, 61, 62, 67, 71, 73, 74, 79, 82, 83, 86, 89, 94, 97, 101, 103, 106, 107, 109, 113, 118, 122, 127, 131, 134, 137, 139, 142, 146, 149, 151, 157, 158, 163, 166

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

For n > 1, a(n) is position of primes in A026741.

For n > 1, a(n) is the position of the ones in A046079. - Ant King, Jan 29 2011

A251561(a(n)) != a(n). - Reinhard Zumkeller, Dec 27 2014

Number of terms <= n is pi(n) + pi(n/2). - Robert G. Wilson v, Aug 04 2017

Number of terms <=10^k: 7, 40, 263, 1898, 14725, 120036, 1013092, 8762589, 77203401, 690006734, 6237709391, 56916048160, 523357198488, 4843865515369, ..., . - Robert G. Wilson v, Aug 04 2017

Complement of A264828. - Chai Wah Wu, Oct 17 2024

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Union of A001747 and A000040.

Subsequence of A039698 and of A033948.

Cf. A026741, A046079, A178156, A251561, A264828.

Programs

Haskell

a001751 n = a001751_list !! (n-1)
a001751_list = 2 : filter (\n -> (a010051 $ div n $ gcd 2 n) == 1) [1..]
-- Reinhard Zumkeller, Jun 20 2011 (corrected, improved), Dec 17 2010

Mathematica

Select[Range[163], Or[PrimeQ[#], PrimeQ[1/2 #]] &] (* Ant King, Jan 29 2011 *)
upto=200;With[{pr=Prime[Range[PrimePi[upto]]]},Select[Sort[Join[pr,2pr]],# <= upto&]] (* Harvey P. Dale, Sep 23 2014 *)

PARI

isA001751(n)=isprime(n/gcd(n,2)) || n==2

PARI

list(lim)=vecsort(concat(primes(primepi(lim)), 2* primes(primepi(lim\2)))) \\ Charles R Greathouse IV, Oct 31 2012

Python

from sympy import primepi
def A001751(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-primepi(x)-primepi(x>>1))
    return bisection(f,n,n) # Chai Wah Wu, Oct 17 2024