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%I #23 Apr 22 2024 14:29:15
%S 1729,14676481,84350561,90698401,279377281,382536001,413138881,
%T 542497201,702683101,781347841,851703301,939947009,955134181,
%U 3480174001,4765950001,5255104513,5781222721,5985964801,7558388641,7816642561,8714965001,9237473281,13630072501,18189007201,21669076801,21863001601,23915494401,25477682491
%N Carmichael numbers k for which bigomega(phi(k)) < bigomega(k-1), where bigomega gives the number of prime divisors, counted with multiplicity.
%C 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.
%H Amiram Eldar, <a href="/A339909/b339909.txt">Table of n, a(n) for n = 1..10000</a> (calculated using data from Claude Goutier)
%H Claude Goutier, <a href="http://www-labs.iro.umontreal.ca/~goutier/OEIS/A055553/">Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22</a>.
%H D. H. Lehmer, <a href="http://dx.doi.org/10.1090/s0002-9904-1932-05521-5">On Euler's totient function</a>, Bulletin of the American Mathematical Society, 38 (1932), 745-751.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer's_totient_problem">Lehmer's totient problem</a>.
%H <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%t carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {_, _}][[;; , 2]]; Select[carmichaels, PrimeOmega[EulerPhi[#]] < PrimeOmega[# - 1] &] (* _Amiram Eldar_, Dec 26 2020 *)
%o (PARI)
%o A002322(n) = lcm(znstar(n)[2]); \\ From A002322
%o isA339909(n) = ((n>1)&&issquarefree(n)&&!isprime(n)&&(bigomega(eulerphi(n))<bigomega(n-1))&&(0==((n-1)%A002322(n))));
%Y Intersection of A002997 and A339908.
%Y Cf. A000010, A001222, A002322.
%Y Cf. also A339818, A339869, A339878.
%K nonn
%O 1,1
%A _Antti Karttunen_, Dec 22 2020
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