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A106092 Even numbers and primes.

%I A106092 #24 Sep 01 2024 20:35:43
%S A106092 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,
%T A106092 32,34,36,37,38,40,41,42,43,44,46,47,48,50,52,53,54,56,58,59,60,61,62,
%U A106092 64,66,67,68,70,71,72,73,74,76,78,79,80,82,83,84,86,88,89,90,92,94,96,97
%N A106092 Even numbers and primes.
%H A106092 Robert Israel, <a href="/A106092/b106092.txt">Table of n, a(n) for n = 1..10000</a>
%H A106092 Paulo A. Martin, <a href="https://doi.org/10.1016/j.jpaa.2003.10.028">The Galois group of x^n-x^(n-1)-...-x-1</a>, Journal of Pure and Applied Algebra, Volume 190, Issues 1-3, 1 June 2004, Pages 213-223.
%F A106092 From _Robert Israel_, Jan 01 2019: (Start)
%F A106092 a(k-1+A000720(2*k)) = 2*k.
%F A106092 a((prime(k)-3)/2 + k) = prime(k) for k >= 2. (End)
%p A106092 map(op,[seq([2*k,`if`(isprime(2*k+1),2*k+1,NULL)],k=1..100)]); # _Robert Israel_, Jan 01 2019
%t A106092 Union[Prime[Range[30]], Select[Range[113], EvenQ]]
%t A106092 DeleteCases[Range[2,100],_?(OddQ[#]&&CompositeQ[#]&)] (* _Harvey P. Dale_, Dec 04 2021 *)
%o A106092 (PARI) isok(n) = isprime(n) || ((n%2) == 0); \\ _Michel Marcus_, Feb 24 2020
%o A106092 (Python)
%o A106092 from sympy import primepi
%o A106092 def A106092(n):
%o A106092     m, k = n, 1+(n<<1)-primepi(n)-(n>>1)
%o A106092     while m != k:
%o A106092         m, k = k, 1+n+k-primepi(k)-(k>>1)
%o A106092     return m # _Chai Wah Wu_, Jul 31 2024
%Y A106092 Cf. A000720.
%Y A106092 Complement of A014076.
%K A106092 nonn,easy
%O A106092 1,1
%A A106092 _Zak Seidov_, May 07 2005