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Numbers k such that k*10^3 + 333 is prime.
2

%I #30 Feb 16 2025 08:32:55

%S 2,5,7,10,16,17,19,20,23,29,31,38,41,49,50,55,56,59,61,64,71,76,79,85,

%T 92,100,101,103,121,134,136,139,140,143,149,154,155,161,175,176,178,

%U 182,184,188,208,209,211,217,220,232,236,239,241,244,265,266,269,271,272,274,286,287,295,299,301,308

%N Numbers k such that k*10^3 + 333 is prime.

%C 10^3 and 333 are relatively prime, therefore by Dirichlet's theorem there are infinitely many primes in the arithmetic progression n*10^3+333. No term of the sequence is of the form 3*k, because 3*k*10^3+333 = 3*(k*10^3+111) is divisible by 3, violating the requirement of the definition. - Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 27 2009

%H Harvey P. Dale, <a href="/A102339/b102339.txt">Table of n, a(n) for n = 1..1000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DirichletsTheorem.html">Dirichlet's Theorem</a>

%e If k=2, then k*10^3 + 333 = 2333 (prime).

%e If k=49, then k*10^3 + 333 = 49333 (prime).

%e If k=92, then k*10^3 + 333 = 92333 (prime).

%t Select[Range[400],PrimeQ[FromDigits[Join[IntegerDigits[#],{3,3,3}]]]&] (* _Harvey P. Dale_, Oct 14 2014 *)

%t Select[Range[0, 1000], PrimeQ[1000 # + 333] &] (* _Vincenzo Librandi_, Jan 19 2013 *)

%o (Magma) [ n: n in [1..700] | IsPrime(Seqint([3,3,3] cat Intseq(n))) ]; // _Vincenzo Librandi_, Feb 04 2011

%o (Magma) [ n: n in [0..320] | IsPrime(n*10^3+333) ]; // _Klaus Brockhaus_, May 20 2009

%o (PARI) is(n)=isprime(1000*n+333) \\ _Charles R Greathouse IV_, Jun 06 2017

%Y Cf. A101472, A157772, A102248.

%K nonn,changed

%O 1,1

%A _Parthasarathy Nambi_, Feb 20 2005