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A180044 Let the n-th Carmichael number A002997(n) = p1*p2*...*pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) * (p1*p2*...*pr - 1)^(r-2) / ((p2-1)*...*(pr-1)).

%I A180044 #25 Sep 08 2022 08:45:54
%S A180044 7,23,48,22,47,45,45,21,44,163,2105352,162,43,486266,3157729,9859600,
%T A180044 5110605,161,6146018,280,8225424,9135075,1684,6185169,1363,159,351,
%U A180044 59907600,950,1675,9879408,1358,949,158,95468562,4399220,83722500
%N A180044 Let the n-th Carmichael number A002997(n) = p1*p2*...*pr, where p1 < p2 < ... < pr are primes. Then a(n) = (p1-1) * (p1*p2*...*pr - 1)^(r-2) / ((p2-1)*...*(pr-1)).
%C A180044 a(n) is always an integer as proved at the Alekseyev link.
%C A180044 The conjecture referred to in A162290 was generalized as follows: Let k be an r-factor Carmichael number (p_1 < p_2 < ... < p_r). Then (p_1-1)*(k-1)^(r-2)/((p_2-1)*(p_3-1)*...*(p_r-1)) is an integer. This was proved by _Max Alekseyev_ (see link).
%C A180044 Contains A162290 as a subsequence.
%H A180044 Amiram Eldar, <a href="/A180044/b180044.txt">Table of n, a(n) for n = 1..10000</a>
%H A180044 Max Alekseyev, <a href="http://www.mersenneforum.org/showthread.php?p=55271&amp;postcount=14">Pomerance's proof</a>.
%e A180044 Since A002997(11) = 41041 = 7*11*13*41, we have a(11) = (6*41040^2) / (10*12*40) = 2105352.
%t A180044 lim = 1000001; CarmichaelQ[n_] := Divisible[n - 1, CarmichaelLambda[n]] && ! PrimeQ[n]; cc = Select[Table[k, {k, 561, lim, 2}], CarmichaelQ]; lg = Length[cc]; a[n_] := (c = cc[[n]]; pp = FactorInteger[c][[All, 1]]; r = Length[pp]; (pp[[1]] - 1)*((Times @@ pp - 1)^(r - 2)/ Times @@ (Drop[pp, 1] - 1))); Table[a[n], {n, 1, lg}] (* _Jean-François Alcover_, Sep 28 2011 *)
%o A180044 (Magma) [ (d[1]-1)*(n-1)^(r-2) / &*[ d[i]-1: i in [2..r] ]: n in [3..700000 by 2] | not IsPrime(n) and IsSquarefree(n) and forall(t){x: x in d | (n-1) mod (x-1) eq 0} where r is #d where d is PrimeDivisors(n)]; // _Klaus Brockhaus_, Aug 10 2010
%Y A180044 Cf. A104016, A104017, A002997, A162290.
%K A180044 nonn
%O A180044 1,1
%A A180044 _A.K. Devaraj_, Aug 08 2010
%E A180044 Edited and extended by _Max Alekseyev_ and _Klaus Brockhaus_, Aug 10 2010