A136204 Primes p such that 3p-2 and 3p+2 are primes (see A125272) and its decimal representation ends in 7.

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

Offset

1,1

Comments

Theorem: If in the triple (3n-2,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

Robert Israel, Table of n, a(n) for n = 1..10000

Maple

filter:= proc(n) isprime(n) and isprime(3*n-2) 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

Cf. A136191, A136192, A125272.

Sequence in context: A106064 A282001 A038862 * A139891 A082113 A196597

Adjacent sequences: A136201 A136202 A136203 * A136205 A136206 A136207

Keyword

nonn,base

Author

Carlos Alves, Dec 21 2007

Status

approved