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

3, 7, 37, 211, 271, 307, 331, 337, 601, 727, 1171, 1237, 1297, 1531, 1657, 2221, 2281, 2557, 3037, 3061, 3067, 4261, 4447, 4801, 4951, 5227, 5581, 5851, 6151, 6361, 6691, 6841, 6967, 7621, 7681, 7687, 7867, 8017, 8167, 8191, 8287, 8521, 8527, 8647, 8941

Offset

1,1

Comments

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).

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.

References

R. K. Guy, Unsolved Problems in Number Theory, A13.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

W. R. Alford, Andrew Granville, and Carl Pomerance, There are infinitely many Carmichael numbers, Ann. of Math. (2) 139 (1994), no. 3, 703-722.

Richard Pinch, Carmichael numbers up to 10^18, April 2006.

Richard Pinch, Carmichael numbers up to 10^18, arXiv:math/0604376 [math.NT], 2006.

Example

For n=3, 2n-1 = 5, 3n-2 = 7.
For n=7, 2n-1 = 13, 3n-2 = 19 and 7*13*19 = 1729 (a Carmichael number).
For n=37, 2n-1 = 73, 3n-2 = 109 and 37*73*109 = 294409 (a Carmichael number).

Maple

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:

Mathematica

Select[Prime[Range[1000]], PrimeQ[2*#-1] && PrimeQ[3*#-2]&] (* Vladimir Joseph Stephan Orlovsky, Jan 13 2011 *)

Prog

(Magma) [ n: n in PrimesUpTo(10000) | IsPrime(2*n-1) and IsPrime(3*n-2) ];
(PARI) forprime(p=3, 10^3, isprime(2*p-1) && isprime(3*p-2) && print1(p, ", ")); \\ Joerg Arndt, Nov 29 2014

Crossrefs

Cf. A002476, A002997, A033502.

Sequence in context: A161675 A208809 A086031 * A152560 A162926 A042895

Adjacent sequences: A174731 A174732 A174733 * A174735 A174736 A174737

Keyword

nonn

Author

Michel Lagneau, Mar 28 2010

Extensions

Typo in term corrected by D. S. McNeil, Nov 20 2010

Status

approved