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A161534
The smallest of four consecutive primes where all three gaps are perfect squares.
4
255763, 604441, 651361, 884497, 913063, 1065133, 1320211, 1526191, 2130133, 2376721, 2907727, 2911933, 2974891, 3190597, 3603583, 3690151, 3707497, 3962941, 4209643, 4245643, 4706101, 5057671, 5155567, 5223187, 5260711, 5321191, 5325571, 5410627
OFFSET
1,1
COMMENTS
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.
A gap of 16 is first involved in 2376721 and 4706101, a gap of 64 first in 4245643, 5710531 and 21953641.
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
EXAMPLE
a(2) = 604441, the smallest of the consecutive primes 604441, 604477, 604481, 604517, with gaps of 36, 4 and 36, all perfect squares.
MATHEMATICA
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 *)
Transpose[Select[Partition[Prime[Range[400000]], 4, 1], And@@IntegerQ/@ Sqrt[ Differences[#]]&]][[1]] (* Harvey P. Dale, Mar 24 2014 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Ki Punches, Jun 13 2009
EXTENSIONS
Terms beyond a(6) from R. J. Mathar, Sep 23 2009
STATUS
approved