

A014688


a(n) = nth prime + n.


63



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
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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
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 kth number after n > 2. Move to successive remaining number n = 3. Then k = 3  1 + 0 = 2: delete the kth 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
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].  YaPing Lu, Sep 02 2020


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
YaPing Lu and ShuFang 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) = 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(n1) + A008578(n+1)  A008578(n) + 1 for n >= 2. a(1) = 3, a(n) = a(n1) + A001223(n1) + 1 for n >= 3.  Jaroslav Krizek, Oct 28 2009
a(n) = 2*OR(p,n)  XOR(p,n), for nth prime p.  Gary Detlefs, Oct 26 2013
a(n) = A078916(n)  n.  Zak Seidov, Nov 10 2013


MAPLE

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


MATHEMATICA

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


PROG

(Haskell)
a014688 n = a014688_list !! (n1)
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

Cf. A000040, A078916, A093570, A093571, A076556, A061068, A332086.
Sequence in context: A175489 A289244 A248879 * A167136 A099836 A344010
Adjacent sequences: A014685 A014686 A014687 * A014689 A014690 A014691


KEYWORD

nonn,easy


AUTHOR

Mohammad K. Azarian


EXTENSIONS

More terms from Vasiliy Danilov (danilovv(AT)usa.net), July 1998
Corrected for changes of offsets of A008578 and A158611 by Jaroslav Krizek, Oct 28 2009


STATUS

approved



