%I #17 Jul 22 2021 23:53:59
%S 1,2,3,5,10,11,12,13,14,15,18,19,20,24,28,31,34,36,37,39,40,41,42,45,
%T 46,48,49,57,64,66,67,68,70,72,73,75,78,79,82,83,86,89,90,92,93,95,96,
%U 97,99,100,103,105,108,109,110,116,117,120,121,124,125,126,128
%N Numbers k such that prime(k)*prime(k+1)*prime(k+2) mod prime(k+3) is even.
%C a(971)=325850. No more terms?
%C Almost surely the above conjecture is true, but it is currently hopeless to prove. It would follow from prime gaps being less than about the cube root, but this is not known even under RH. - _Charles R Greathouse IV_, Jun 10 2017
%t Select[Range@ 128, EvenQ@ Mod[Times @@ Take[#, 3], #[[4]]] &@ Prime[# + Range[0, 3]] &] (* _Michael De Vlieger_, Jun 09 2017 *)
%t Position[Partition[Prime[Range[140]],4,1],_?(EvenQ[Mod[Times@@Take[#,3], #[[4]]]]&)]//Flatten//Quiet (* _Harvey P. Dale_, Oct 14 2020 *)
%o (PARI) isok(n) = (((prime(n)*prime(n+1)*prime(n+2)) % prime(n+3)) % 2) == 0; \\ _Michel Marcus_, Jun 07 2017
%o (PARI) list(lim)=my(v=List(),p=2,q=3,r=5,n); forprime(s=7,, if(n++>lim, break); if(p*q*r%s%2==0, listput(v,n)); p=q; q=r; r=s); Vec(v) \\ _Charles R Greathouse IV_, Jun 10 2017
%K nonn
%O 1,2
%A _Zak Seidov_, Jun 06 2017