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A161534 The smallest of four consecutive primes where all three gaps are perfect squares.

%I A161534 #12 Nov 07 2018 21:52:44
%S A161534 255763,604441,651361,884497,913063,1065133,1320211,1526191,2130133,
%T A161534 2376721,2907727,2911933,2974891,3190597,3603583,3690151,3707497,
%U A161534 3962941,4209643,4245643,4706101,5057671,5155567,5223187,5260711,5321191,5325571,5410627
%N A161534 The smallest of four consecutive primes where all three gaps are perfect squares.
%C A161534 Gaps occur as (36,4,36), (4,36,36), etc., all with at least one of them equal to 36 thru primes of 10^9.
%C A161534 A gap of 16 is first involved in 2376721 and 4706101, a gap of 64 first in 4245643, 5710531 and 21953641.
%C A161534 a(26) = 5321191 = A052239(6) = A058252(1) is the first term to be followed by three equal gaps, i.e., to start a sequence of consecutive primes in arithmetic progression (CPAP-4). - _M. F. Hasler_, Nov 06 2018
%H A161534 T. D. Noe, <a href="/A161534/b161534.txt">Table of n, a(n) for n = 1..1000</a>
%e A161534 a(2) = 604441, the smallest of the consecutive primes 604441, 604477, 604481, 604517, with gaps of 36, 4 and 36, all perfect squares.
%t A161534 PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; t = {}; n = 3; p1 = 1; p2 = 2; p3 = 3; p4 = 5; While[Length[t] < 30, n++; p1 = p2; p2 = p3; p3 = p4; p4 = Prime[n]; If[PerfectSquareQ[p2 - p1] && PerfectSquareQ[p3 - p2] && PerfectSquareQ[p4 - p3], AppendTo[t, p1]]]; t (* _T. D. Noe_, Jul 09 2013 *)
%t A161534 Transpose[Select[Partition[Prime[Range[400000]],4,1],And@@IntegerQ/@ Sqrt[ Differences[#]]&]][[1]] (* _Harvey P. Dale_, Mar 24 2014 *)
%Y A161534 Cf. A161002, A161533, A138198.
%K A161534 nonn
%O A161534 1,1
%A A161534 _Ki Punches_, Jun 13 2009
%E A161534 Terms beyond a(6) from _R. J. Mathar_, Sep 23 2009