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A014688 a(n) = n-th prime + n. 55
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 (list; graph; refs; listen; history; text; internal format)
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 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

Complement of A064427. - Jaroslav Krizek, Oct 28 2009

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Carlos Rivera, Puzzle 821. Prime numbers and complementary sequences, Prime Puzzles.

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

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 !! (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

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

Sequence in context: A175489 A248879 * A167136 A099836 A281333 A244031

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

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Last modified May 30 04:27 EDT 2017. Contains 287305 sequences.