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

%I A072225 #46 Aug 13 2025 05:06:07
%S A072225 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,
%T A072225 60,61,62,64,68,69,70,75,77,79,80,83,87,88,90,91,94,95,97,101,113,116,
%U A072225 119,120,121,126,128,132,133,134
%N A072225 Numbers k such that prime(k) + prime(k+1) + prime(k+2) is prime.
%C A072225 The sequence contains 298884 terms <= 2000000, so nearly 15% of the numbers <= 2000000 are in the sequence. - _Dmitry Kamenetsky_, Aug 08 2015
%C A072225 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
%C A072225 The first 8 consecutive integers in this sequence start from 8744076. - _Dmitry Kamenetsky_, Aug 28 2015
%C A072225 The first 9 consecutive integers in this sequence start from 697642916. - _Dmitry Kamenetsky_, Sep 08 2015
%C A072225 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
%H A072225 Robert Israel, <a href="/A072225/b072225.txt">Table of n, a(n) for n = 1..10000</a>
%H A072225 Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_798.htm">Puzzle 798. A nice puzzle by Kamenetski</a>, The Prime Puzzles & Problems Connection.
%e A072225 9 is in the sequence because prime(9) + prime(10) + prime(11) = 23 + 29 + 31 = 83 is a prime.
%p A072225 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
%t A072225 Select[Range[10^4], PrimeQ[Prime[ # ] + Prime[ # + 1] + Prime[ # + 2]] &]
%o A072225 (Magma) [n: n in [0..600]| IsPrime(NthPrime(n)+NthPrime(n+1)+NthPrime(n+2))]; // _Vincenzo Librandi_, Apr 06 2011
%o A072225 (PARI) isok(n)=isprime(prime(n)+prime(n+1)+prime(n+2)) \\ _Anders Hellström_, Aug 20 2015
%Y A072225 Cf. A034962, A073681, A117673, A152469, A152470.
%Y A072225 Cf. A386275 (first occurrences of runs).
%K A072225 easy,nonn
%O A072225 1,1
%A A072225 _Joseph L. Pe_, Jul 04 2002