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A001751 Primes together with primes multiplied by 2.

%I A001751 #40 Oct 17 2024 12:52:16
%S A001751 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,
%T A001751 58,59,61,62,67,71,73,74,79,82,83,86,89,94,97,101,103,106,107,109,113,
%U A001751 118,122,127,131,134,137,139,142,146,149,151,157,158,163,166
%N A001751 Primes together with primes multiplied by 2.
%C A001751 For n > 1, a(n) is position of primes in A026741.
%C A001751 For n > 1, a(n) is the position of the ones in A046079. - _Ant King_, Jan 29 2011
%C A001751 A251561(a(n)) != a(n). - _Reinhard Zumkeller_, Dec 27 2014
%C A001751 Number of terms <= n is pi(n) + pi(n/2). - _Robert G. Wilson v_, Aug 04 2017
%C A001751 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
%C A001751 Complement of A264828. - _Chai Wah Wu_, Oct 17 2024
%H A001751 T. D. Noe, <a href="/A001751/b001751.txt">Table of n, a(n) for n = 1..10000</a>
%t A001751 Select[Range[163], Or[PrimeQ[#], PrimeQ[1/2 #]] &] (* _Ant King_, Jan 29 2011 *)
%t A001751 upto=200;With[{pr=Prime[Range[PrimePi[upto]]]},Select[Sort[Join[pr,2pr]],# <= upto&]] (* _Harvey P. Dale_, Sep 23 2014 *)
%o A001751 (Haskell)
%o A001751 a001751 n = a001751_list !! (n-1)
%o A001751 a001751_list = 2 : filter (\n -> (a010051 $ div n $ gcd 2 n) == 1) [1..]
%o A001751 -- _Reinhard Zumkeller_, Jun 20 2011 (corrected, improved), Dec 17 2010
%o A001751 (PARI) isA001751(n)=isprime(n/gcd(n,2)) || n==2
%o A001751 (PARI) list(lim)=vecsort(concat(primes(primepi(lim)), 2* primes(primepi(lim\2)))) \\ _Charles R Greathouse IV_, Oct 31 2012
%o A001751 (Python)
%o A001751 from sympy import primepi
%o A001751 def A001751(n):
%o A001751     def bisection(f,kmin=0,kmax=1):
%o A001751         while f(kmax) > kmax: kmax <<= 1
%o A001751         while kmax-kmin > 1:
%o A001751             kmid = kmax+kmin>>1
%o A001751             if f(kmid) <= kmid:
%o A001751                 kmax = kmid
%o A001751             else:
%o A001751                 kmin = kmid
%o A001751         return kmax
%o A001751     def f(x): return int(n+x-primepi(x)-primepi(x>>1))
%o A001751     return bisection(f,n,n) # _Chai Wah Wu_, Oct 17 2024
%Y A001751 Union of A001747 and A000040.
%Y A001751 Subsequence of A039698 and of A033948.
%Y A001751 Cf. A026741, A046079, A178156, A251561, A264828.
%K A001751 nonn,easy
%O A001751 1,1
%A A001751 _N. J. A. Sloane_