A072225 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is prime.

3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 18, 19, 20, 22, 23, 26, 29, 34, 35, 37, 38, 45, 47, 50, 58, 60, 61, 62, 64, 68, 69, 70, 75, 77, 79, 80, 83, 87, 88, 90, 91, 94, 95, 97, 101, 113, 116, 119, 120, 121, 126, 128, 132, 133, 134

Offset: 1

Author

Joseph L. Pe, Jul 04 2002

Keywords

• easy
• nonn

Comments

The sequence contains 298884 terms <= 2000000, so nearly 15% of the numbers <= 2000000 are in the sequence. - Dmitry Kamenetsky, Aug 08 2015

Heuristically, we might expect the "probability" of n being in the sequence to be on the order of 1/log(n). - Robert Israel, Aug 09 2015

The first 8 consecutive integers in this sequence start from 8744076. - Dmitry Kamenetsky, Aug 28 2015

The first 9 consecutive integers in this sequence start from 697642916. - Dmitry Kamenetsky, Sep 08 2015

Vladimir Chirkov found that the first 10 and 11 consecutive integers in this sequence start from 23169509240 and 29165083170, respectively (see The Prime Puzzles link and A386275). - Dmitry Kamenetsky, Sep 08 2015

Examples

			9 is in the sequence because prime(9) + prime(10) + prime(11) = 23 + 29 + 31 = 83 is a prime.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Carlos Rivera, Puzzle 798. A nice puzzle by Kamenetski, The Prime Puzzles & Problems Connection.

Crossrefs

Cf. A034962, A073681, A117673, A152469, A152470.

Cf. A386275 (first occurrences of runs).

Programs

Magma

[n: n in [0..600]| IsPrime(NthPrime(n)+NthPrime(n+1)+NthPrime(n+2))]; // Vincenzo Librandi, Apr 06 2011

Maple

a:=proc(n) if isprime(ithprime(n)+ithprime(n+1)+ithprime(n+2))=true then n else fi end: seq(a(n),n=1..150); # Emeric Deutsch, Apr 24 2006

Mathematica

Select[Range[10^4], PrimeQ[Prime[ # ] + Prime[ # + 1] + Prime[ # + 2]] &]

PARI

isok(n)=isprime(prime(n)+prime(n+1)+prime(n+2)) \\ Anders Hellström, Aug 20 2015