A106092 Even numbers and primes.

2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 50, 52, 53, 54, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 70, 71, 72, 73, 74, 76, 78, 79, 80, 82, 83, 84, 86, 88, 89, 90, 92, 94, 96, 97

Offset: 1

Author

Zak Seidov, May 07 2005

Keywords

• nonn
• easy

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Paulo A. Martin, The Galois group of x^n-x^(n-1)-...-x-1, Journal of Pure and Applied Algebra, Volume 190, Issues 1-3, 1 June 2004, Pages 213-223.

Crossrefs

Cf. A000720.

Complement of A014076.

Programs

Maple

map(op,[seq([2*k,`if`(isprime(2*k+1),2*k+1,NULL)],k=1..100)]); # Robert Israel, Jan 01 2019

Mathematica

Union[Prime[Range[30]], Select[Range[113], EvenQ]]
DeleteCases[Range[2,100],?(OddQ[#]&&CompositeQ[#]&)] (* _Harvey P. Dale, Dec 04 2021 *)

PARI

isok(n) = isprime(n) || ((n%2) == 0); \\ Michel Marcus, Feb 24 2020

Python

from sympy import primepi
def A106092(n):
    m, k = n, 1+(n<<1)-primepi(n)-(n>>1)
    while m != k:
        m, k = k, 1+n+k-primepi(k)-(k>>1)
    return m # Chai Wah Wu, Jul 31 2024

Formula

From Robert Israel, Jan 01 2019: (Start)

a(k-1+A000720(2*k)) = 2*k.

a((prime(k)-3)/2 + k) = prime(k) for k >= 2. (End)