A276980 Odd squarefree numbers n > 1 such that lambda(n)^2 = phi(n), where lambda is the Carmichael lambda function and phi is Euler's totient function.

273, 1729, 2109, 2255, 4433, 4641, 4697, 5673, 6643, 6935, 7667, 8103, 8729, 10235, 11543, 14497, 16385, 16523, 17507, 18915, 20033, 22649, 23579, 26691, 29309, 29393, 34799, 35853, 35929, 37209, 37829, 39277, 42653, 45551, 55699, 56163, 68735, 68901, 69167, 69977, 70993, 73505, 75361, 76373

Offset

1,1

Comments

Such a number n must have at least three prime factors.

Are there infinitely many such numbers?

Among them are some Carmichael numbers: 1729, 75361, ... (A002997).

Links

Robert G. Wilson v, Table of n, a(n) for n = 1..1024

Example

273 = 3 * 7 * 13, so phi(273) = 2 * 6 * 12 = 144 = 12^2 and lambda(273) = lcm(2, 6, 12) = 12, hence 273 is in the sequence.
Notice that phi(315) = 144 and lambda(315) = 12 also. However, mu(315) = 0 since 315 = 3^2 * 5 * 7, so for that reason 315 is not in the sequence.

Mathematica

samePsiSqPhiQ[n_] := SquareFreeQ[n] && CarmichaelLambda[n]^2 == EulerPhi[n]; Select[1 + 2 Range@50000, samePsiSqPhiQ] (* Robert G. Wilson v, Apr 14 2017 *)

Prog

(PARI) is(n) = n>1 && n%2!=0 && issquarefree(n) && lcm(znstar(n)[2])^2==eulerphi(n) \\ Felix Fröhlich, Apr 22 2017

Crossrefs

Subsequence of A024556.

Cf. A000010, A002322, A002997.

Sequence in context: A043467 A028530 A200836 * A350303 A214220 A029567

Adjacent sequences: A276977 A276978 A276979 * A276981 A276982 A276983

Keyword

nonn

Author

Thomas Ordowski and Altug Alkan, Apr 11 2017

Status

approved