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%I #7 Sep 17 2017 22:50:59
%S 561,1729,41041,1615681,14676481,40622401,173085121,367804801,
%T 631071001,8494657921,138399075361,432081216001,997803878401,
%U 3837165696001,7599525373441,42182344790209,65032633451521,186137387251201,329797704600001,2523853463040001
%N Carmichael numbers c with record number of primes p such that c*p is also a Carmichael number.
%C If c*p is a Carmichael number, where p is a prime, then (p-1)|(c-1), so given c, the number of possible primes is bounded by the number of divisors of c-1.
%C The corresponding number of solutions is 0, 5, 7, 10, 12, 14, 18, 26, 30, 33, 55, 65, 71, 72, 90, 92, 112, 128, 192, 218.
%e 1729 has 5 prime numbers p: 37, 73, 109, 433 and 577, such that 1729*p: 63973, 126217, 188461, 748657 and 997633 are also Carmichael numbers.
%t carmichaelQ[n_] := Not[PrimeQ[n]] && Divisible[n - 1, CarmichaelLambda[n]];
%t numSol[n_] := Module[{m = 0}, ds = Divisors[n-1]; Do[p = ds[[k]] + 1; If[! PrimeQ[p], Continue[]]; If[!carmichaelQ[p*n], Continue[]]; m++, {k, 1, Length[ds] - 1}]; m]; numSolmax=-1;seq={}; Do[n=A002997[[j]];m=numSol[n];If[m>numSolmax,AppendTo[seq,n];numSolmax=m],{j,1,Length[A002997]}];seq
%Y Cf. A002997.
%K nonn
%O 1,1
%A _Amiram Eldar_, Sep 15 2017
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