

A133313


Primes p such that 3p2 and 3p+2 are primes (see A125272) and its decimal representation finishes with 3.


0



3, 13, 23, 43, 103, 163, 293, 313, 433, 523, 953, 1013, 1063, 1153, 1283, 1303, 1483, 1693, 1723, 1783, 1913, 2003, 2333, 3533, 3823, 3943, 4003, 4013, 4093, 4943, 5483, 6043, 6133, 6173, 6473, 6803, 7523, 7573, 7603, 7673, 7853, 7993, 8513, 9283, 9343
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OFFSET

1,1


COMMENTS

Theorem: If in the triple (3n2,n,3n+2) all numbers are primes, then n=5 or the decimal representation of n finishes with 3 or 7. Proof: Similar to A136191. Alternative Mathematica proof: Table[nn = 10k + r; Intersection @@ (Divisors[CoefficientList[(3nn  2) nn(3nn + 2), k]]), {r, 1, 9, 2}]; This gives {{1, 5}, {1}, {1, 5}, {1}, {1, 5}}. Therefore only r=3 and r=7 allow nontrivial divisors (excluding nn=5 itself).


LINKS

Table of n, a(n) for n=1..45.


MATHEMATICA

TPrimeQ = (PrimeQ[ #  2] && PrimeQ[ #/3] && PrimeQ[ # + 2]) &; Select[Select[Range[100000], TPrimeQ]/3, Mod[ #, 10] == 3 &]
Select[Prime[Range[1200]], Mod[#, 10]==3&&AllTrue[3#+{2, 2}, PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 18 2019 *)


CROSSREFS

Cf. A136204 (finishing with 7), A136191, A136192, A125272.
Sequence in context: A147473 A030431 A090146 * A230026 A260798 A102010
Adjacent sequences: A133310 A133311 A133312 * A133314 A133315 A133316


KEYWORD

nonn,base


AUTHOR

Carlos Alves, Dec 21 2007


STATUS

approved



