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 A014688 a(n) = n-th prime + n. 56

%I

%S 3,5,8,11,16,19,24,27,32,39,42,49,54,57,62,69,76,79,86,91,94,101,106,

%T 113,122,127,130,135,138,143,158,163,170,173,184,187,194,201,206,213,

%U 220,223,234,237,242,245,258,271,276,279,284,291,294,305,312,319,326

%N a(n) = n-th prime + n.

%C Conjecture: this sequence contains an infinite number of primes (A061068), yet contains arbitrarily long "prime deserts" such as the 11 composites in A014688 between a(6) = 19 and a(18) = 79 and the 17 composites in A014688 between a(48) = 271 and a(66) = 383. - _Jonathan Vos Post_, Nov 22 2004

%C Does an n exist such that n*prime(n)/(n+prime(n)) is an integer? - _Ctibor O. Zizka_, Mar 04 2008. The answer to Zizka's question is easily seen to be No: such an integer k would be positive and less than prime(n), but then k*(n + prime(n)) = prime(n)*n would be impossible. - _Robert Israel_, Apr 20 2015

%C May be obtained by a sieve on the sequence of natural numbers. Starting from n=1 delete the number corresponding to the alternate sum of the preceding left numbers. Iterate with the successive left number. First step n = 1, k = 1 - 0 = 1: delete the k-th number after n -> 2. Move to successive remaining number n = 3. Then k = 3 - 1 + 0 = 2: delete the k-th number after n -> 5. Move to successive remaining number n = 4. Then k = 4 - 3 + 1 - 0 = 2. After 4 we have 6, 7, 8, ... (5 deleted in previous step). So delete n = 7. And so on. - _Paolo P. Lava_ and _Giorgio Balzarotti_, Jul 14 2008

%C Complement of A064427. - _Jaroslav Krizek_, Oct 28 2009

%H Reinhard Zumkeller, <a href="/A014688/b014688.txt">Table of n, a(n) for n = 1..10000</a>

%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_821.htm">Puzzle 821. Prime numbers and complementary sequences</a>, Prime Puzzles.

%F a(n) = n + A000040(n) = n + A008578(n+1) = n + A158611(n+2). - _Jaroslav Krizek_, Aug 31 2009

%F a(n) = A090178(n+1) - 1 = (n+1)-th noncomposite number + n for n >= 2. a(n) = A167136(n+1). a(1) = 3, a(n) = a(n-1) + A008578(n+1) - A008578(n) + 1 for n >= 2. a(1) = 3, a(n) = a(n-1) + A001223(n-1) + 1 for n >= 3. - _Jaroslav Krizek_, Oct 28 2009

%F a(n) = 2*OR(p,n) - XOR(p,n), for n-th prime p. - _Gary Detlefs_, Oct 26 2013

%F a(n) = A078916(n) - n. - _Zak Seidov_, Nov 10 2013

%p P:=proc(i) local a,n; for n from 1 by 1 to i do a:=ithprime(n)+n; print(a); od; end: P(100); # _Paolo P. Lava_, Jul 14 2008

%t Table[n + Prime[n], {n, 100}] (* _T. D. Noe_, Dec 06 2012 *)

%o a014688 n = a014688_list !! (n-1)

%o a014688_list = zipWith (+) [1..] a000040_list

%o -- _Reinhard Zumkeller_, Sep 16 2011

%o (PARI) a(n)=prime(n)+n \\ _Charles R Greathouse IV_, Mar 21 2013

%o (MAGMA) [NthPrime(n)+n: n in [1..70]]; // _Vincenzo Librandi_ Jan 02 2016

%Y Cf. A000040, A078916, A093570, A093571, A076556, A061068.

%K nonn,easy

%O 1,1