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

3, 5, 8, 11, 16, 19, 24, 27, 32, 39, 42, 49, 54, 57, 62, 69, 76, 79, 86, 91, 94, 101, 106, 113, 122, 127, 130, 135, 138, 143, 158, 163, 170, 173, 184, 187, 194, 201, 206, 213, 220, 223, 234, 237, 242, 245, 258, 271, 276, 279, 284, 291, 294, 305, 312, 319, 326

Offset: 1

Author

Mohammad K. Azarian

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000040, A078916, A093570, A093571, A076556, A061068, A332086.

See also A065995.

Programs

Haskell

a014688 n = a014688_list !! (n-1)
a014688_list = zipWith (+) [1..] a000040_list
-- Reinhard Zumkeller, Sep 16 2011

Magma

[NthPrime(n)+n: n in [1..70]]; // Vincenzo Librandi Jan 02 2016

Mathematica

Table[n + Prime[n], {n, 100}] (* T. D. Noe, Dec 06 2012 *)

PARI

a(n)=prime(n)+n \\ Charles R Greathouse IV, Mar 21 2013

Formula

a(n) = n + A000040(n) = n + A008578(n+1) = n + A158611(n+2). - Jaroslav Krizek, Aug 31 2009

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

Extensions

More terms from Vasiliy Danilov (danilovv(AT)usa.net), Jul 1998

Corrected for changes of offsets of A008578 and A158611 by Jaroslav Krizek, Oct 28 2009