All code combined

NeverDividesLucasSeqQ[a_,p_]:=And@@Table[Mod[a[n],p]>0,{n,0,p-1}];

A2[a_,b_,c_,n_/;n<0]=0;
A2[a_,b_,c_,0]=1;
A2[a_,b_,c_,n_]:=A2[a,b,c,n]=((a (n-1)^2+a (n-1)+b)A2[a,b,c,n-1]-c(n-1)^2A2[a,b,c,n-2])/n^2;
A3[a_,b_,c_,n_/;n<0]=0;
A3[a_,b_,c_,0]=1;
A3[a_,b_,c_,n_]:=A3[a,b,c,n]=(((2n-1)(a(n-1)^2+a(n-1)+b))A3[a,b,c,n-1]-c(n-1)^3A3[a,b,c,n-2])/n^3;
A3[a_,b_,c_,d_,n_/;n<0]=0;
A4[a_,b_,c_,d_,n_]:=A4[a,b,c,d,n]=(((2n-1)(a(n-1)^2+a(n-1)+b))A4[a,b,c,d,n-1]-(n-1)(c(n-1)^2+d)A4[a,b,c,d,n-2])/n^3;
A4[a_,b_,c_,d_,n_/;n<0]=0;
A4[a_,b_,c_,d_,0]=1;
Aa[n_]:=A2[7,2,-8,n]
Ab[n_]:=A2[11,3,-1,n]
Ac[n_]:=A2[10,3,9,n]
Ad[n_]:=A2[12,4,32,n]
Af[n_]:=A2[9,3,27,n]
Ag[n_]:=A2[17,6,72,n]
Aδ[n_]:=A3[7,3,81,n] 
Aη[n_]:=A3[11,5,125,n]  
Aα[n_]:=A3[10,4,64,n]
Aϵ[n_]:=A3[12,4,16,n]
Aζ[n_]:=A3[9,3,-27,n]
Aγ[n_]:=A3[17,5,1,n]
As7[n_]:=A4[13,4,-27,3,n]
As10[n_]:=A4[6,2,-64,4,n]
As18[n_]:=A4[14,6,192,-12,n]

Code just for sequence B (A005258), as an example

A2[a_,b_,c_,n_/;n<0]=0;
A2[a_,b_,c_,0]=1;
A2[a_,b_,c_,n_]:=A2[a,b,c,n]=((a (n-1)^2+a (n-1)+b)A2[a,b,c,n-1]-c(n-1)^2A2[a,b,c,n-2])/n^2;
Ab[n_]:=A2[11,3,-1,n];
NeverDividesLucasSeqQ[a_,p_]:=And@@Table[Mod[a[n],p]>0,{n,0,p-1}];
Select[Range[100],PrimeQ[#]&&NeverDividesLucasSeqQ[Ab,#]&]

The primes up to 10,000 for each sequence

A000172:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Aa,#]&]

A005258:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Ab,#]&]

A002893:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Ac,#]&]

A081085:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Ad,#]&]

A006077:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Af,#]&]

A093388:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Ag,#]&]

A125143: (* upto sign *)

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Aδ,#]&]

A229111: (*up to sign: *)

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Aη,#]&]

A002895:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Aα,#]&]

A290575:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Aϵ,#]&]

A290576:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Aζ,#]&]

A005259:

Select[Range[10000],PrimeQ[#]&&NeverDividesLucasSeqQ[Aγ,#]&]

A005260: s10 
{}-- All primes divide at some stage

A219692:s18

{}-- All primes divide at some stage

A183204: s7
{}--All primes divide at some stage