%I #12 Sep 08 2022 08:45:05

%S 2,26,34,68,124,160,188,342,602,776,3104,6324,14688,17170,35894,94500,

%T 97094,111036,113102,122180,138096,150314,150624,195396,270106,496706,

%U 1035380,1318064,1428542,1445120,1968392,2015720,3149874,3185300,3774572,4466898,4970816

%N Numbers n such that determinant[{{n,phi(n)},{n+1,phi(n+1)}}]is a perfect square.

%C If n is a term of the sequence, then the parallelogram formed by the vectors {n,phi(n)},{n+1,phi(n+1)} has the same area as that of an integral square.

%e Det[{{26,phi(26)},{27,phi(27)}}] = Det[{{26,12},{27,18}}] = 12^2, so 26 is a term of the sequence.

%t f[n_] := Det[{{n, EulerPhi[n]}, {n + 1, EulerPhi[n + 1]}}]; Do[If[IntegerQ @ Sqrt @ f[n], Print[n]], {n, 1, 10^5}]

%o (PARI) isok(n) = issquare(matdet([n, eulerphi(n); n+1, eulerphi(n+1)])); \\ _Michel Marcus_, Sep 26 2019

%o (Magma) [k:k in [1..5000000]|IsSquare(k*EulerPhi(k+1)-(k+1)*EulerPhi(k))]; // _Marius A. Burtea_, Sep 26 2019

%Y Cf. A000010, A067564, A067572.

%K nonn

%O 1,1

%A _Joseph L. Pe_, Jan 30 2002

%E a(18)-a(37) from _Amiram Eldar_, Sep 26 2019