A067317 Numbers k such that 1 + binomial(k,j) is prime for only 2 values of j (0 <= j <= k).

1, 3, 7, 15, 23, 31, 59, 63, 67, 81, 84, 93, 95, 127, 157, 170, 214, 239, 253, 255, 313, 470, 511, 622, 694, 1010, 1023, 1098, 1691, 2047, 3535, 3836, 3963, 4095, 6143, 7166, 8191, 11757, 12525, 12686, 16383, 32767

Offset: 1

Author

Labos Elemer, Jan 15 2002

Keywords

• nonn
• more

Examples

			The 2 values of j are 0 and n, which give the prime 2. The sequence includes all numbers of the form 2^m-1 since binomial(2^m-1,j) is odd for all j.
		

Crossrefs

Cf. A067316.

Programs

Mathematica

test[n_] := Module[{}, For[i=1, 2i<=n, i++, If[PrimeQ[Binomial[n, i]+1], Return[False]]]; True]; For[n=1, True, n++, If[test[n], Print[n]]]

PARI

isok(n) = sum(j=0, n, isprime(1 + binomial(n,j))) == 2; \\ Michel Marcus, Oct 30 2018

PARI

is(n) = if(n == 1, 1, for(i=1, n\2, if(isprime(binomial(n, i) + 1), return(0))); 1); \\ Amiram Eldar, Jul 18 2024

Formula

Numbers k such that A067316(k) = 2.

Extensions

More terms from Jon E. Schoenfield, May 30 2010

a(35)-a(41) from Robert Israel, Mar 09 2020

a(42) from Amiram Eldar, Jul 18 2024