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A174734 Prime numbers n such that 2n-1 and 3n-2 are prime.

%I A174734 #39 Sep 08 2022 08:45:51
%S A174734 3,7,37,211,271,307,331,337,601,727,1171,1237,1297,1531,1657,2221,
%T A174734 2281,2557,3037,3061,3067,4261,4447,4801,4951,5227,5581,5851,6151,
%U A174734 6361,6691,6841,6967,7621,7681,7687,7867,8017,8167,8191,8287,8521,8527,8647,8941
%N A174734 Prime numbers n such that 2n-1 and 3n-2 are prime.
%C A174734 If n, 2n-1 and 3n-2 are prime numbers, and if n >= 5, then n*(2*n-1)*(3*n-2) is a Carmichael number (A033502).
%C A174734 Proof: there exist numbers m such that n=6m+1 is prime (if n=6m+5, then 2n-1 = 12m+9 is composite). Let p=(6m+1)(12m+1)(18m+1) = a*b*c. Then p-1 = 6*12*18*m^3 + (6*12 + 6*18 + 12*18)*m^2 + (6 + 12 + 19)*m, so p-1 is divisible by a-1=6m, by b-1=12m, and by c-1=18m; thus p is a Carmichael number.
%D A174734 R. K. Guy, Unsolved Problems in Number Theory, A13.
%H A174734 Charles R Greathouse IV, <a href="/A174734/b174734.txt">Table of n, a(n) for n = 1..10000</a>
%H A174734 W. R. Alford, Andrew Granville, and Carl Pomerance, <a href="http://www.jstor.org/stable/2118576">There are infinitely many Carmichael numbers</a>, Ann. of Math. (2) 139 (1994), no. 3, 703-722.
%H A174734 Richard Pinch, <a href="http://www.chalcedon.demon.co.uk/rgep/publish.html#76">Carmichael numbers up to 10^18</a>, April 2006.
%H A174734 Richard Pinch, <a href="http://arxiv.org/abs/math/0604376">Carmichael numbers up to 10^18</a>, arXiv:math/0604376 [math.NT], 2006.
%e A174734 For n=3, 2n-1 = 5, 3n-2 = 7.
%e A174734 For n=7, 2n-1 = 13, 3n-2 = 19 and 7*13*19 = 1729 (a Carmichael number).
%e A174734 For n=37, 2n-1 = 73, 3n-2 = 109 and 37*73*109 = 294409 (a Carmichael number).
%p A174734 with(numtheory): for n from 2 to 15000 do: if type(n,prime)=true and type(2*n-1,prime)=true and type(3*n-2,prime)=true then print (n):else fi:od:
%t A174734 Select[Prime[Range[1000]], PrimeQ[2*#-1] && PrimeQ[3*#-2]&] (* _Vladimir Joseph Stephan Orlovsky_, Jan 13 2011 *)
%o A174734 (Magma) [ n: n in PrimesUpTo(10000) | IsPrime(2*n-1) and IsPrime(3*n-2) ];
%o A174734 (PARI) forprime(p=3,10^3, isprime(2*p-1) && isprime(3*p-2) && print1(p,", ")); \\ _Joerg Arndt_, Nov 29 2014
%Y A174734 Cf. A002476, A002997, A033502.
%K A174734 nonn
%O A174734 1,1
%A A174734 _Michel Lagneau_, Mar 28 2010
%E A174734 Typo in term corrected by _D. S. McNeil_, Nov 20 2010