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A293620 Numbers k such that f(k), f(k+1) and f(k+2) are all primes, where f(k) = (2k+1)^2 - 2 (A073577).

%I A293620 #32 Jun 02 2025 15:25:58
%S A293620 1,2,16,58,149,177,534,681,954,1045,1052,1255,1367,1563,2046,2074,
%T A293620 2515,2557,2564,2788,3586,3593,3908,4062,4552,5252,5371,5385,6400,
%U A293620 6729,7443,7478,9305,9375,9942,10355,10411,10726,10740,11286,11545,11559,11832,11965
%N A293620 Numbers k such that f(k), f(k+1) and f(k+2) are all primes, where f(k) = (2k+1)^2 - 2 (A073577).
%C A293620 Sierpiński proved that under Schinzel's hypothesis H this sequence is infinite.
%C A293620 Sierpiński showed that the only quadruple of consecutive primes of the form (2k+1)^2 - 2 are for k = 1 (i.e., 1 and 2 are the only consecutive terms in this sequence).
%C A293620 Numbers k such that the 3 consecutive integers k, k+1 and k+2 belong to A088572. - _Michel Marcus_, Oct 13 2017
%H A293620 Amiram Eldar, <a href="/A293620/b293620.txt">Table of n, a(n) for n = 1..10000</a>
%H A293620 Wacław Sierpiński, <a href="http://gdz.sub.uni-goettingen.de/en/dms/load/img/?PPN=PPN311571026_0017&amp;DMDID=dmdlog25">Remarque sur la distribution de nombres premiers</a>, Matematički Vesnik, Vol. 2(17), Issue 31 (1965), pp. 77-78.
%H A293620 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime</a>.
%H A293620 Wikipedia, <a href="https://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H">Schinzel's hypothesis H</a>.
%e A293620 The first triples are: k = 1: (7, 23, 47), k = 2: (23, 47, 79), k = 16: (1087, 1223, 1367).
%t A293620 Select[Range[10^4], AllTrue[{(2#+1)^2-2, (2#+3)^2-2, (2#+5)^2-2},PrimeQ] &]
%t A293620 SequencePosition[Table[If[PrimeQ[(2k+1)^2-2],1,0],{k,12000}],{1,1,1}][[;;,1]] (* _Harvey P. Dale_, Feb 09 2025 *)
%o A293620 (PARI) f(n) = 4*n^2 + 4*n - 1;
%o A293620 isok(n) = isprime(f(n)) && isprime(f(n+1)) && isprime(f(n+2)); \\ _Michel Marcus_, Oct 13 2017
%Y A293620 Cf. A088572, A008865, A028870, A028871, A073577, A088572.
%K A293620 nonn
%O A293620 1,2
%A A293620 _Amiram Eldar_, Oct 13 2017