

A136192


Primes p such that 2p3 and 2p+3 are both prime (A092110), with last decimal of p being 7.


5



7, 17, 67, 97, 127, 137, 157, 167, 487, 547, 617, 647, 937, 1187, 1277, 1427, 1627, 1847, 2027, 2297, 2437, 2467, 2477, 2617, 2857, 2927, 3137, 3457, 3727, 4007, 4057, 4157, 5167, 5417, 5657, 6247, 6257, 7027, 7477, 7867, 8467, 8737, 8747, 9127, 9227
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OFFSET

1,1


COMMENTS

Except for p=5, the decimals in A092110 end in 3 or 7.
Theorem: If in the triple (2n3,n,2n+3) all numbers are primes then n=5 or the decimal representation of n ends in 3 or 7. Proof: Consider Q=(2n3)n(2n+3), by hypothesis factorized into primes. If n is prime, n=10k+r with r=1,3,7 or 9. We want to exclude r=1 and r=9. Case n=10k+1. Then Q=5(1+6k+240k^2+800k^3) and 5 is a factor; thus 2n3=5 or n=5 or 2n+1=5 : this means n=4 (not prime); or n=5 (included); or n=2 (impossible, because 2n3=1). Case n=10k+9. Then Q=5(567+1926k+2160k^2+800k^3) and 5 is a factor; the arguments, for the previous case, also hold.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..1000


MATHEMATICA

bpQ[n_]:=Last[IntegerDigits[n]]==7&&And@@PrimeQ[2n+{3, 3}]; Select[Prime[ Range[1200]], bpQ] (* Harvey P. Dale, Sep 25 2013 *)


CROSSREFS

Cf. A092110, A136191.
Sequence in context: A276907 A293464 A106010 * A269239 A118431 A051809
Adjacent sequences: A136189 A136190 A136191 * A136193 A136194 A136195


KEYWORD

nonn,base


AUTHOR

Carlos Alves, Dec 20 2007


EXTENSIONS

Definition clarified by Harvey P. Dale, Sep 25 2013


STATUS

approved



