A290805 Least Carmichael number whose Euler totient function value is an n-th power.

561, 1729, 63973, 1729, 367939585, 63973, 294409, 232289960085001, 11570858964626401, 79939760257, 509033161, 611559276803883001, 13079177569, 27685385948423487745, 26979791457662785, 287290964059686145, 13046319747121261903830001, 7847507962539316696504321, 993942550111105, 6280552422566791778305, 24283361157780097, 759608966313690599499265, 6657107145346817668085761

Offset

1,1

Comments

Banks proved that for each positive integer N there are an infinite number of Carmichael numbers whose Euler totient function value is an N-th power. Therefore this sequence is infinite.

The terms were calculated using Pinch's tables of Carmichael numbers (see link below).

a(25) = 33420122657338444417, a(26) = 239468866473584181889, and there are no more terms below 10^22. - Amiram Eldar, Apr 20 2024

Links

Table of n, a(n) for n=1..23.

William D. Banks, Carmichael Numbers with a Square Totient, Canadian Mathematical Bulletin, Vol. 52, No. 1 (2009), pp. 3-8.

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.

R. G. E. Pinch, Tables relating to Carmichael numbers.

Example

phi(1729) = 36^2 = 6^4 while phi(561) and phi(1105) are not perfect powers, therefore a(2) = a(4) = 1729.

Crossrefs

Cf. A000010, A002997, A272798.

Sequence in context: A224930 A083732 A292367 * A217465 A097130 A110889

Adjacent sequences: A290802 A290803 A290804 * A290806 A290807 A290808

Keyword

nonn

Author

Amiram Eldar, Aug 11 2017

Extensions

a(1) prepended by David A. Corneth, Aug 11 2017

a(14)-a(16), a(19)-a(21) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024

a(17)-a(18), a(22)-a(23) from Max Alekseyev, Apr 25 2024

Status

approved