%I #15 Apr 16 2023 04:12:26
%S 1,2136,13494274080,216818853118725
%N Let k = A000396(n) be the n-th perfect number, a(n) is the least number m such that k*d*m + 1 is prime for all of the proper divisors d of k so their product is a Carmichael number.
%C Chernick proved that (6m + 1)*(12m + 1)*(18m + 1) is a Carmichael number, if all the 3 factors are primes (A033502, A046025).
%C Lieuwens generalized it to Product_{i} (k*d(i)*m + 1), for k a perfect number.
%C a(1) corresponds to 6. It was found by Jack Chernick in 1939.
%C a(2) corresponds to 28. It was found by Dubner in 1996. Lieuwens evaluated that the least corresponding Carmichael number > 10^27.
%C a(3) corresponds to 496. It was found by Jim Fougeron in 2002 (Dubner found a larger value: 474382033125).
%C a(4) corresponds to 8128. It was found by _Phil Carmody_ in 2002.
%C The corresponding Carmichael numbers are 1729, 599966117492747584686619009, 1.631... * 10^126, 4.559... * 10^260, ...
%D Harold Davenport, The Higher Arithmetic, Cambridge University Press, 7th ed., 1999, exercise 8.4.
%D Harvey Dubner, Carmichael numbers and Egyptian fractions, Mathematica japonicae, Vol. 43, No. 2 (1996), pp. 411-419.
%H Jack Chernick, <a href="http://projecteuclid.org/euclid.bams/1183501763">On Fermat's simple theorem</a>, Bulletin of the American Mathematical Society, Vol. 45, No. 4 (1939), pp. 269-274.
%H Claude Goutier, <a href="https://cms.math.ca/publications/crux/issue/?volume=46&issue=8">De l'utilité d'une vieille curiosité grecque</a>, Crux Mathematicorum, Vol. 46, No. 8 (2020), pp. 397-403.
%H G. L. Honaker, Jr. and Chris Caldwell, <a href="https://t5k.org/curios/page.php?number_id=2602">Prime Curios! 59996...19009 (27-digits)</a>.
%H Erik Lieuwens, <a href="https://repository.tudelft.nl/islandora/object/uuid:becf731b-0b3c-4b2f-8479-9d4f205659d9?collection=research">Fermat pseudo primes</a>, Doctoral Thesis, Delft University of Technology, 1971, pp. 29-30.
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_171.htm">Puzzle 171. Perfect & Carmichael numbers</a>, The Prime Puzzles & Problems Connection.
%e 28 = 1 + 2 + 4 + 7 + 14 is the second perfect number. 2136 is the least number m such that 28*1*333 + 1 = 59809, 28*2*2136 + 1 = 119617, 28*4*2136 + 1 = 239233, 28*7*2136 + 1 = 418657 and 28*14*2136 + 1 = 837313 are all primes, therefore 59809*119617*239233*418657*837313 = 599966117492747584686619009 is a Carmichael number.
%t ms = {2, 3, 5, 7, 13}; ns = Length[ms]; M[p_] := 2^(p - 1)*(2^p - 1); L[m_] := Module[{}, d = Most[Divisors[m]]*m; aQ[n_] := AllTrue[d*n + 1, PrimeQ]; n=1; While[!aQ[n], n++];n]; s={}; Do[m = M[ms[[k]]]; b = L[m]; AppendTo[s, b], {k, 1, ns}]; s
%Y Cf. A000396, A002997, A033502, A046025, A067199.
%K nonn,bref,more
%O 1,2
%A _Amiram Eldar_, Sep 07 2018