1,1

There are no more terms < 10^6. Triangles A(k), A(k+1), A(k+2) with an obtuse angle at A(k+1) seem to be less frequent than those with an acute angle, but also for large k (e.g. 999998) now and then an obtuse angle at A(k+1) occurs. So this line of thought does not provide an argument against the existence of further terms.

To verify that 5 is a term: A(5) = [4,11], A(6) = [2,13], A(7) = [6,17]; square of leg A(5)--A(6) is (4-2)^2+(11-13)^2 = 4+4 = 8; square of leg A(6)--A(7) is (2-6)^2+(13-17)^2 = 16+16 = 32, square of leg A(5)--A(7) is (4-6)^2+(11-17)^2 =

4+36 = 40; now 8 + 32 = 40 and the angle at A(6) is a right one.

p[ n_ ] := EulerPhi[ n ]; s[ n_ ] := Prime[ n ]; d[ x1_, y1_, x2_, y2_ ] := (x1 - x2)^2 + (y1 - y2)^2; Select[ Range[ 1, 10^6 ], d[ p[ # ], s[ # ], p[ # + 1 ], s[ # + 1 ] ] + d[ p[ # + 1 ], s[ # + 1 ], p[ # + 2 ], s[ # + 2 ] ] == d[ p[ # ], s[ # ], p[ # + 2 ], s[ # + 2 ] ] & ]

(PARI) {for(j=1, 1000000, x0=eulerphi(j); x1=eulerphi(j+1); x2=eulerphi(j+2); y0=prime(j); y1=prime(j+1); y2=prime(j+2); k1=norml2([x0, y0]-[x1, y1]); k2=norml2([x1, y1]-[x2, y2]); h=norml2([x0, y0]-[x2, y2]); if(k1+k2==h, print1(j, ", ")))}

Sequence in context: A124519 A154311 A014980 this_sequence A019125 A019205 A042958

Adjacent sequences: A066898 A066899 A066900 this_sequence A066902 A066903 A066904

nonn

Joseph L. Pe (joseph_l_pe(AT)hotmail.com), Jan 22 2002

Edited by Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 03 2003