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Numbers k such that prime(k) * prime(k+1) mod prime(k+2) is odd.
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%I #31 Jan 16 2023 14:49:04

%S 1,2,5,7,10,14,15,23,29,46,61

%N Numbers k such that prime(k) * prime(k+1) mod prime(k+2) is odd.

%C Finite? Full?

%C Next term, if it exists, is greater than 1026351685.

%C From _Robert Israel_, Jun 19 2017: (Start)

%C Numbers k such that floor(A001223(k+1)*A031131(k)/prime(k+2)) is odd.

%C Cramér's conjecture implies the sequence is finite. (End)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture">Cramér's conjecture</a>

%e The first five primes are 2, 3, 5, 7, 11.

%e We see that 2 * 3 = 1 mod 5, so 1 (corresponding to the first prime, 2) is in the sequence.

%e We see that 3 * 5 = 1 mod 7, so 2 (corresponding to the second prime, 3) is in the sequence.

%e But 5 * 7 = 2 mod 11, so 3 (corresponding to the third prime, 5) is not in the sequence.

%p P:= select(isprime, [2,seq(i,i=3..10^6,2)]):

%p select(n -> (P[n]*P[n+1] mod P[n+2])::odd, [$1..nops(P)-2]); # _Robert Israel_, Jun 19 2017

%t Select[Range[1000], OddQ[Mod[Prime[#] Prime[# + 1], Prime[# + 2]]] &] (* _Alonso del Arte_, Jun 06 2017 *)

%t Position[Partition[Prime[Range[70]],3,1],_?(OddQ[Mod[#[[1]]#[[2]], #[[3]]]]&),1,Heads->False]//Flatten (* _Harvey P. Dale_, Aug 11 2017 *)

%o (PARI) isok(n) = (((prime(n) * prime(n + 1)) % prime(n + 2)) % 2); \\ _Michel Marcus_, Jun 07 2017

%Y Cf. A001223, A006094, A031131, A182126.

%K nonn,more

%O 1,2

%A _Zak Seidov_, Jun 06 2017