%I #15 Sep 21 2015 20:37:26

%S 189029,404471,424663,2595221,4140901,4197377,4347209,4528159,4566193,

%T 4631023,4708819,4864411,5175589,5311729,6380651,6400819,6426029,

%U 7117783,8173877,8915971,10080589,10460869,10671173,11094661,11538313

%N Quasi-Carmichael numbers to exactly six bases.

%e a(1) = 189029 because this is the first squarefree composite number n such that exactly six integers b except 0 exist such that for every prime factor p of n, p+b divides n+b (-419, -414, -407, -389, -365, -309): 189029=421*449 and 2, 30 both divide 188610 and 7, 35 both divide 188615 and 14, 42 both divide 188622 and 32, 60 both divide 188640 and 56, 84 both divide 188664 and 112, 140 both divide 188720.

%o (PARI) for(n=2, 1000000, if(!isprime(n), if(issquarefree(n), f=factor(n); k=0; for(b=-(f[1, 1]-1), n, c=0; for(i=1, #f[, 1], if((n+b)%(f[i, 1]+b)>0, c++)); if(c==0, if(!b==0, k++))); if(k==6, print1(n, ", ")))))

%Y Cf. A257750 (every number of bases).

%Y Cf. A257751, A257752, A257753, A257754, A257755, A257757, A258842 (1, 2, 3, 4, 5, 7 and 8 bases).

%Y Cf. A257758 (first occurrences).

%K nonn

%O 1,1

%A _Tim Johannes Ohrtmann_, May 12 2015