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A339869 Carmichael numbers k for which A053575(k) [the odd part of phi] divides k-1.

%I A339869 #32 Apr 22 2024 14:28:01
%S A339869 561,1105,2465,6601,8911,10585,46657,62745,162401,410041,449065,
%T A339869 5148001,5632705,6313681,6840001,7207201,11119105,11921001,19683001,
%U A339869 21584305,26719701,41298985,55462177,64774081,67371265,79411201,83966401,87318001,99861985,100427041,172290241,189941761,484662529,790623289,809883361
%N A339869 Carmichael numbers k for which A053575(k) [the odd part of phi] divides k-1.
%C A339869 Lehmer conjectured that the equation k * phi(n) = n - 1 (with k integer) has no solutions for any composite n (i.e., when k > 1). If this sequence has no common terms with A339818, then the conjecture certainly holds.
%H A339869 Amiram Eldar, <a href="/A339869/b339869.txt">Table of n, a(n) for n = 1..1150</a> (terms below 10^22 calculated using data from Claude Goutier)
%H A339869 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 A339869 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 A339869 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer&#39;s_totient_problem">Lehmer's totient problem</a>.
%H A339869 <a href="/index/Ca#Carmichael">Index entries for sequences related to Carmichael numbers</a>.
%t A339869 carmichaels = Cases[Import["https://oeis.org/A002997/b002997.txt", "Table"], {_, _}][[;; , 2]]; oddPart[n_] := n/2^IntegerExponent[n, 2]; q[n_] := Divisible[n - 1, oddPart[EulerPhi[n]]]; Select[carmichaels, q] (* _Amiram Eldar_, Dec 26 2020 *)
%o A339869 (PARI)
%o A339869 A000265(n) = (n>>valuation(n, 2));
%o A339869 A002322(n) = lcm(znstar(n)[2]);
%o A339869 isA339869(n) = ((n>1)&&!isprime(n)&&(!((n-1)%A002322(n)))&&!((n-1)%A000265(eulerphi(n))));
%Y A339869 Intersection of A002997 and A339880.
%Y A339869 Complement of A340092 in A002997.
%Y A339869 Cf. A000010, A000265, A002322, A053575.
%Y A339869 Cf. also A339818, A339878, A339909.
%K A339869 nonn
%O A339869 1,1
%A A339869 _Antti Karttunen_, Dec 22 2020