

A136204


Primes p such that 3p2 and 3p+2 are primes (see A125272) and its decimal representation ends in 7.


2



7, 37, 127, 167, 257, 337, 757, 797, 887, 1307, 1597, 1657, 1667, 2347, 2557, 2897, 2927, 3067, 4297, 4327, 4877, 5087, 5147, 5227, 5417, 5857, 6337, 6827, 6917, 6967, 7127, 7187, 7547, 7687, 7867, 7877, 8147, 8447, 8527, 8647, 9857, 10037, 10687
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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 ends in 3 or 7. Proof: Similar to A136191. Alternative Mathematica proof: Table[nn = 10k + r; Intersection (AT)(AT) (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



MAPLE

filter:= proc(n) isprime(n) and isprime(3*n2) and isprime(3*n+2) end proc:
select(filter, [seq(i, i=7..10^4, 10)]); # Robert Israel, Nov 20 2023


MATHEMATICA

TPrimeQ = (PrimeQ[ #  2] && PrimeQ[ #/3] && PrimeQ[ # + 2]) &; Select[Select[Range[100000], TPrimeQ]/3, Mod[ #, 10] == 7 &]


CROSSREFS



KEYWORD

nonn,base


AUTHOR



STATUS

approved



