A066901 Let phi(k) determine the x-coordinate and prime(k) the y-coordinate of the point A(k). Sequence gives values of n such that points A(n), A(n+1) and A(n+2) form a right triangle with right angle at A(n+1).

5, 6, 9, 21, 44

Offset

1,1

Comments

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.

There are no more terms < 10^8. - Sean A. Irvine, Nov 24 2023

Links

Table of n, a(n) for n=1..5.

Example

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.

Mathematica

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 ] ] & ]

Prog

(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, ", ")))}

Crossrefs

Cf. A000010, A000040.

Sequence in context: A154311 A279702 A014980 * A273039 A019125 A019205

Adjacent sequences: A066898 A066899 A066900 * A066902 A066903 A066904

Keyword

nonn,more

Author

Joseph L. Pe, Jan 22 2002

Extensions

Edited by Klaus Brockhaus, Jun 03 2003

Status

approved