A339878 Carmichael numbers k such that phi(k) divides p*(k - 1) for some prime factor p of k - 1.

1729, 3069196417, 23915494401, 1334063001601, 6767608320001, 33812972024833, 1584348087168001, 1602991137369601, 6166793784729601, 1531757211193440001, 84388996672599528001

Offset: 1

Author

Antti Karttunen (after Thomas Ordowski's and Amiram Eldar's SeqFan-posting), Dec 26 2020

Keywords

• nonn
• more

Comments

The first ten terms are all in A339818, none is in A339869, and all except a(2) and a(6) are in A339909.

Also, for all ten, a(n) == 1 (mod 64). (Cf. a similar comment in A338998).

Links

• Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
• Thomas Ordowski and Amiram Eldar, A new look at the Lehmer's totient problem, SeqFan, February 10 2019.

Crossrefs

Intersection of A002997 and A338998.

Cf. also A339818, A339869, A339909.

Programs

Mathematica

carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {, }][[;; , 2]]; q[n_] := Module[{p = FactorInteger[n - 1][[;; , 1]], phi = EulerPhi[n]}, AnyTrue[(n - 1)*p, Divisible[#, phi] &]]; Select[carmichaels, q] (* Amiram Eldar, Dec 26 2020 *)

Extensions

a(10) from Amiram Eldar, Dec 26 2020

a(11) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 21 2024