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A161533 The smallest of three consecutive primes p1 < p2 < p3, where p2-p1, p3-p2, and p3-p1 are all perfect squares. 4
623071, 779377, 1744891, 2055853, 2906887, 3168721, 3540793, 4177573, 4245643, 4245679, 4309957, 4449127, 4833271, 4858981, 5541187, 5550583, 5710531, 5710567, 5856931, 6013591, 6789637, 6855493, 7024627, 7162339, 7340383, 7614847, 8143501 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Note that sqrt(p2-p1), sqrt(p3-p2), sqrt(p3-p1) form a Pythagorean triple. [corrected by James R. Buddenhagen, Jul 09 2013]

Gap pairs p1-p2, p3-p2 occur as 36,64, or 64,36 at least through a(n) <= 10^8.

LINKS

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

EXAMPLE

623071 is the smallest of the consecutive primes 623071, 623107, and 623171 with gaps 623107-623071 = 36, 623171-623107 = 64, and the double gap 623171-623071 = 100 each a perfect square.

MATHEMATICA

PerfectSquareQ[n_] := JacobiSymbol[n, 13] =!= -1 && JacobiSymbol[n, 19] =!= -1 && JacobiSymbol[n, 17] =!= -1 && JacobiSymbol[n, 23] =!= -1 && IntegerQ[Sqrt[n]]; t = {}; n = 2; p1 = 1; p2 = 2; p3 = 3; While[Length[t] < 30, n++; p1 = p2; p2 = p3; p3 = Prime[n]; If[PerfectSquareQ[p2 - p1] && PerfectSquareQ[p3 - p2] && PerfectSquareQ[p3 - p1], AppendTo[t, p1]]]; t (* T. D. Noe, Jul 09 2013 *)

psQ[{a_, b_, c_}]:=AllTrue[{Sqrt[b-a], Sqrt[c-b], Sqrt[c-a]}, IntegerQ]; Transpose[ Select[Partition[Prime[Range[600000]], 3, 1], psQ]][[1]] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Aug 20 2014 *)

CROSSREFS

Cf. A161002, A138198.

Sequence in context: A341892 A236064 A204052 * A174975 A053877 A236008

Adjacent sequences:  A161530 A161531 A161532 * A161534 A161535 A161536

KEYWORD

nonn

AUTHOR

Ki Punches, Jun 13 2009

EXTENSIONS

5710567 inserted by R. J. Mathar, Sep 23 2009

STATUS

approved

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Last modified October 16 23:04 EDT 2021. Contains 348047 sequences. (Running on oeis4.)