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%I A289337 #6 Jul 18 2017 12:12:14
%S A289337 25,125,325,451,1561,4089,7107,8625,12025
%N A289337 Composite numbers (pseudoprimes) n, that are not Carmichael numbers, such that A000670(n-1) == 0 (mod n).
%C A289337 I. J. Good proved that A000670(k*(p-1)) == 0 (mod p) for all k >= 1 and prime p. Therefore the congruence A000670(n-1) == 0 (mod n) holds for all primes and Carmichael numbers. This sequence consist of the other composite numbers for which the congruence holds.
%H A289337 I. J. Good, <a href="http://www.fq.math.ca/Scanned/13-1/good.pdf">The number of orderings of n candidates when ties are permitted</a>, Fibonacci Quarterly, Vol. 13 (1975), pp. 11-18.
%e A289337 A000670(24) = 2958279121074145472650648875 is divisible by 25 and 25 is not a prime, nor a Carmichael number.
%t A289337 a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k]*a[n - k], {k, 1, n}]; carmichaelQ[n_]:=(Mod[n, CarmichaelLambda[n]] == 1); seqQ[n_] := !carmichaelQ[n] && Divisible[a[n-1],n]; Select[Range[2,500],seqQ]
%Y A289337 Cf. A000670, A002997, A289338.
%K A289337 nonn,more
%O A289337 1,1
%A A289337 _Amiram Eldar_, Jul 02 2017

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