A257759 - OEIS

%I #10 Aug 27 2015 10:55:43

%S 1105,1595,2093,2465,2821,7843,10373,17963,19721,29341,31003,33143,

%T 46189,46657,62647,66263,70151,70219,88559,101813,106361,115843,

%U 193343,200777,206471,209933,230159,234883,252601,285619,294409,308267,343027,369799,423181,467273

%N Quasi-Carmichael numbers to at least one negative base and at least one positive base.

%C It is a open question whether any Carmichael number exists that is also a Lucas-Carmichael number.

%H Tim Johannes Ohrtmann, <a href="/A257759/b257759.txt">Table of n, a(n) for n = 1..130</a>

%e a(1) = 1105 because this is the first squarefree composite number n such that at least one negative integer and at least one positive integer except 0 exist such that for every prime factor p of n applies that p+b divides n+b (-1, 15): 1105=5*13*17 and 4, 12, 16 both divide 1104 and 20, 28, 32 both divide 1120.

%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, if(k==0, k++), if(b>0, if(k==1, k++))))); if(k==2, print1(n,", ")))))

%Y Intersection of A259282 and A259283.

%K nonn

%O 1,1

%A _Tim Johannes Ohrtmann_, May 12 2015