The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A174734 Prime numbers n such that 2n-1 and 3n-2 are prime. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 13 05:35 EDT 2024. Contains 373366 sequences. (Running on oeis4.)