%I #37 Jul 22 2019 08:40:57
%S 561,1105,46657,52633,188461,670033,825265,838201,1082809,2455921,
%T 2628073,4463641,4767841,5632705,8830801,11119105,13187665,16778881,
%U 18307381,18900973,21584305,22665505,31146661,31405501,31692805,34657141,36765901,38624041,40280065
%N Carmichael numbers m such that A309132(m) < m.
%C A309132(m) divides m for all Carmichael numbers m, but apparently most of them equal A309132(m). Of the first 10000 Carmichael numbers, only 1341 are in this sequence.
%C The ratios a(n)/A309132(a(n)) are 3, 5, 13, 7, 133, 7, 133, 7, 7, 793, 7, 13, 13, ...
%C By _Jonathan Sondow_'s theorem (cf. comments in A309132), these are Carmichael numbers m such that denominator(Sum_{prime p|m}1/p - 1/m) < m, i.e., A326690(m) < m.
%C Problem: are there Carmichael numbers m such that A309132(m) is prime? Equivalently, Carmichael numbers m such that A326690(m) is prime. None exist below 2^64. Conjecture: there are no such Carmichael numbers.
%H Amiram Eldar, <a href="/A309268/b309268.txt">Table of n, a(n) for n = 1..10000</a>
%t aQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]] && Denominator[ Total@(1/FactorInteger[n][[;; , 1]]) - 1/n] < n; Select[Range[10^6], aQ]
%Y Cf. A309132, A326690.
%Y Subsequence of A002997 and A309235.
%K nonn
%O 1,1
%A _Amiram Eldar_ and _Thomas Ordowski_, Jul 20 2019