A334906 - OEIS

%I #20 Jul 02 2021 01:59:52

%S 1,2,6,7,12,19,20,26,33,34,37,38,43,44,45,56,60,63,68,71,75,78,82,83,

%T 86,89,94,95,106,112,115,116,122,135,140,141,142,148,151,152,166,169,

%U 175,178,197,198,206,210,211,226,227,233,236,244,251,264,285,286,287,288,301,302,313,314,321,322

%N Numbers k such that binomial(prime(k+2), prime(k+1)) and binomial(prime(k+1), prime(k)) are coprime.

%C Numbers k such that A058078(k)=1.

%C If prime(k+1)-prime(k)=2 and prime(k+2)-prime(k+1)=4, then k is in the sequence unless prime(k) == 3 (mod 8) or prime(k) == 5 (mod 9).

%C If prime(k+1)-prime(k)=4 and prime(k+2)-prime(k+1)=2, then k is in the sequence unless prime(k) == 7 (mod 8) or prime(k) == 7 (mod 9).

%H Robert Israel, <a href="/A334906/b334906.txt">Table of n, a(n) for n = 1..10000</a>

%e a(3)=6 is in the sequence because the 6th, 7th and 8th primes are 13, 17 and 19, and binomial(17,13)=2380 and binomial(19,17)=171 are coprime.

%p filter:= n -> igcd(binomial(ithprime(n+2),ithprime(n+1)),binomial(ithprime(n+1),ithprime(n)))=1:

%p select(filter, [$1..1000]);

%o (PARI) isok(k) = gcd(binomial(prime(k+2), prime(k+1)), binomial(prime(k+1), prime(k))) == 1; \\ _Michel Marcus_, Jul 02 2021

%Y Cf. A058078.

%K nonn

%O 1,2

%A _Robert Israel_, May 15 2020