A339909 Carmichael numbers k for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.

1729, 14676481, 84350561, 90698401, 279377281, 382536001, 413138881, 542497201, 702683101, 781347841, 851703301, 939947009, 955134181, 3480174001, 4765950001, 5255104513, 5781222721, 5985964801, 7558388641, 7816642561, 8714965001, 9237473281, 13630072501, 18189007201, 21669076801, 21863001601, 23915494401, 25477682491

Offset

1,1

Comments

Natural numbers n that satisfy equation k * phi(n) = n - 1, for some integer k > 1, should all occur in this sequence, if they exist at all. Lehmer conjectured that there are no such numbers.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000 (calculated using data from Claude Goutier)

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

D. H. Lehmer, On Euler's totient function, Bulletin of the American Mathematical Society, 38 (1932), 745-751.

Wikipedia, Lehmer's totient problem.

Index entries for sequences related to Carmichael numbers.

Mathematica

carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {_, _}][[;; , 2]]; Select[carmichaels, PrimeOmega[EulerPhi[#]] < PrimeOmega[# - 1] &] (* Amiram Eldar, Dec 26 2020 *)

Prog

(PARI)
A002322(n) = lcm(znstar(n)[2]); \\ From A002322
isA339909(n) = ((n>1)&&issquarefree(n)&&!isprime(n)&&(bigomega(eulerphi(n))<bigomega(n-1))&&(0==((n-1)%A002322(n))));

Crossrefs

Intersection of A002997 and A339908.

Cf. A000010, A001222, A002322.

Cf. also A339818, A339869, A339878.

Sequence in context: A138130 A306657 A048949 * A339878 A258166 A130876

Adjacent sequences: A339906 A339907 A339908 * A339910 A339911 A339912

Keyword

nonn,changed

Author

Antti Karttunen, Dec 22 2020

Status

approved