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, 283219223388059484626764342346640001

Offset: 1

Author

Amiram Eldar, Aug 11 2017

Keywords

• nonn

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.

For any n > 26, a(n) > 10^22. - Amiram Eldar, Apr 20 2024

Examples

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

Links

• Max Alekseyev, Table of n, a(n) for n = 1..60
• 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.

Crossrefs

Cf. A000010, A002997, A272798, A292573.

Extensions

Terms up to a(13) were calculated using Pinch's tables of Carmichael numbers.

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

a(14)-a(16), a(19)-a(21), a(25)-a(26) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 20 2024

a(17)-a(18), a(22)-a(24) from Max Alekseyev, Apr 25 2024

Edited by Max Alekseyev, Dec 04 2024