OFFSET
1,1
COMMENTS
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
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
Complement of A064427. - Jaroslav Krizek, Oct 28 2009
According to a theorem of Lu and Deng (see LINKS), there exists at least one prime number p such that a(n)-n < p <= a(n); equivalently pi(a(n)) - pi(a(n)-n) >= 1 (see A332086). For example, prime number 3 is in the range of (2,3], 5 in (3,5], 7 in (5,8], and 29 & 31 in (23,32]. - Ya-Ping Lu, Sep 02 2020
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ya-Ping Lu and Shu-Fang Deng, An upper bound for the prime gap, arXiv:2007.15282 [math.GM], 2020.
Carlos Rivera, Puzzle 821. Prime numbers and complementary sequences, The Prime Puzzles and Problems Connection.
Juan Luis Varona, On the Solution of the Equation n = a*k + b*p_k by Means of an Iterative Method, Journal of Integer Sequences, Vol. 24 (2021), Article 21.10.5.
FORMULA
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
a(n) = 2*OR(p,n) - XOR(p,n), for n-th prime p. - Gary Detlefs, Oct 26 2013
a(n) = A078916(n) - n. - Zak Seidov, Nov 10 2013
MATHEMATICA
Table[n + Prime[n], {n, 100}] (* T. D. Noe, Dec 06 2012 *)
PROG
(Haskell)
a014688 n = a014688_list !! (n-1)
a014688_list = zipWith (+) [1..] a000040_list
-- Reinhard Zumkeller, Sep 16 2011
(PARI) a(n)=prime(n)+n \\ Charles R Greathouse IV, Mar 21 2013
(Magma) [NthPrime(n)+n: n in [1..70]]; // Vincenzo Librandi Jan 02 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Vasiliy Danilov (danilovv(AT)usa.net), July 1998
STATUS
approved