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

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

Author

Ki Punches, Jun 13 2009

Keywords

• nonn

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

Examples

			a(2) = 604441, the smallest of the consecutive primes 604441, 604477, 604481, 604517, with gaps of 36, 4 and 36, all perfect squares.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A161002, A161533, A138198.

Programs

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 *)

Extensions

Terms beyond a(6) from R. J. Mathar, Sep 23 2009