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

1,2

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).

Links

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

Example

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.

Maple

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

Prog

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

Crossrefs

Cf. A058078.

Sequence in context: A105329 A124319 A325262 * A078471 A127406 A092310

Adjacent sequences: A334903 A334904 A334905 * A334907 A334908 A334909

Keyword

nonn

Author

Robert Israel, May 15 2020

Status

approved