A334906 Numbers k such that binomial(prime(k+2), prime(k+1)) and binomial(prime(k+1), prime(k)) are coprime.

1, 2, 6, 7, 12, 19, 20, 26, 33, 34, 37, 38, 43, 44, 45, 56, 60, 63, 68, 71, 75, 78, 82, 83, 86, 89, 94, 95, 106, 112, 115, 116, 122, 135, 140, 141, 142, 148, 151, 152, 166, 169, 175, 178, 197, 198, 206, 210, 211, 226, 227, 233, 236, 244, 251, 264, 285, 286, 287, 288, 301, 302, 313, 314, 321, 322

Offset: 1

Author

Robert Israel, May 15 2020

Keywords

• nonn

Comments

Numbers k such that A058078(k)=1.

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).

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).

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A058078.

Programs

Maple

filter:= n -> igcd(binomial(ithprime(n+2),ithprime(n+1)),binomial(ithprime(n+1),ithprime(n)))=1:
select(filter, [$1..1000]);

PARI

isok(k) = gcd(binomial(prime(k+2), prime(k+1)), binomial(prime(k+1), prime(k))) == 1; \\ Michel Marcus, Jul 02 2021