A257759 Quasi-Carmichael numbers to at least one negative base and at least one positive base.

1105, 1595, 2093, 2465, 2821, 7843, 10373, 17963, 19721, 29341, 31003, 33143, 46189, 46657, 62647, 66263, 70151, 70219, 88559, 101813, 106361, 115843, 193343, 200777, 206471, 209933, 230159, 234883, 252601, 285619, 294409, 308267, 343027, 369799, 423181, 467273

Offset: 1

Author

Tim Johannes Ohrtmann, May 12 2015

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..130

Crossrefs

Intersection of A259282 and A259283.

Programs

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,", ")))))