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A294064 Numbers k such that 2*k - 3, 2*k + 3, 3*k - 2, 3*k + 2 are primes.

%I A294064 #34 Mar 03 2024 10:16:15
%S A294064 5,7,13,35,43,55,77,127,133,155,167,253,287,295,365,475,497,533,595,
%T A294064 713,1007,1177,1483,1805,2323,2575,2723,2927,3107,3415,3487,3823,4145,
%U A294064 4213,4367,4565,4717,4927,4963,5125,5215,5363,5417,5587,5627,5795,6133,6587,6797
%N A294064 Numbers k such that 2*k - 3, 2*k + 3, 3*k - 2, 3*k + 2 are primes.
%C A294064 The common numbers of A098090, A067076, A153183, A024893.
%C A294064 Conjecture: The Sum_{n>=1} 1/a(n) = 0.57... converges.
%C A294064 Note that the sum of the 4 primes that are obtained is 10 times the original term: (2*k - 3) + (2*k + 3) + (3*k - 2) + (3*k + 2) = 10*k.
%C A294064 From _Robert G. Wilson v_, Nov 19 2017: (Start)
%C A294064 Number of terms less than 10^m: 2, 7, 20, 55, 189, 919, 4863, 28218, 174469, ..., ;
%C A294064 Number of prime terms less than 10^m: 2, 4, 6, 12, 39, 140, 558, 2755, 14804, ..., .
%C A294064 All terms are == {5, 7, 13, 17, 23, 25} (mod 30).
%C A294064 (End)
%H A294064 Robert G. Wilson v, <a href="/A294064/b294064.txt">Table of n, a(n) for n = 1..10000</a>
%e A294064 5 is in the sequence because 2*5-3 = 7, 2*5+3 = 13, 3*5-2 = 13, 3*5+2 = 17 and the tetrad [7, 13, 13, 17] are all prime numbers.
%e A294064 7 is in the sequence because 2*7-3 = 11, 2*7+3 = 17, 3*7-2 = 19, 3*7+2 = 23 and the tetrad [11, 17, 19, 23] are all prime numbers.
%t A294064 Select[Range[10^4], Function[k, AllTrue[Flatten@ Map[#1 k + {-1, 1} #2 & @@ # &, {#, Reverse@ #}] &@ {2, 3}, PrimeQ]]] (* _Michael De Vlieger_, Oct 22 2017 *)
%o A294064 (PARI) {
%o A294064 for(n=1,10000,
%o A294064     if(isprime(2*n-3)&&isprime(2*n+3)&&isprime(3*n-2)&&isprime(3*n+2),
%o A294064        print1(n", ")
%o A294064       )
%o A294064    )
%o A294064 }
%Y A294064 Cf. A098090, A067076, A153183, A024893.
%K A294064 nonn
%O A294064 1,1
%A A294064 _Dimitris Valianatos_, Oct 22 2017