A085918 Primes p such that for some k the number of terms > 0 and < 1 in the Farey sequence of order k is p.

3, 5, 11, 17, 31, 41, 71, 79, 101, 127, 139, 149, 179, 199, 211, 229, 241, 269, 277, 307, 359, 383, 431, 449, 541, 773, 829, 881, 1259, 1307, 1327, 1493, 1831, 1933, 2141, 2551, 3373, 3947, 4127, 4831, 4957, 5021, 5153, 5323, 5431, 5569, 5813, 6091, 6329

Offset: 1

Author

Cino Hilliard, Aug 16 2003

Keywords

• easy
• nonn

Comments

Or, prime numbers of the form Sum_{j=2..k} phi(j). - Jorge Coveiro, Dec 22 2004. Examples: phi(2)+phi(3) = 3; phi(2)+phi(3)+phi(4) = 5; phi(2)+phi(3)+phi(4)+phi(5)+phi(6) = 11; phi(2)+phi(3)+phi(4)+phi(5)+phi(6)+phi(7) = 17.

Does this sequence have an infinite number of terms?

Examples

			The Farey sequence of order 4 is {0, 1/4, 1/3, 1/2, 2/3, 3/4, 1}. The number of terms > 0 and < 1 is 5, which is prime, so 5 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A078334, A005728, A101300, A000010.

Programs

Mathematica

Select[Accumulate[Table[EulerPhi[k], {k, 2, 150}]], PrimeQ] (* Amiram Eldar, Jul 06 2024 *)

PARI

/* Farey sequence of order n */ fareycountp(n) = { for(x=2,n, y = farey(x); if(isprime(y),print1(y",")); ) }
farey(n) = { c=1; m=n*(n-2)+2; a=vector(m); for(x=1,n, for(y=x,n, v = x/y; if(v<1, c++; a[c]=v; ) ) ); a = vecsort(a); c=0; for(x=2,m, if(a[x]<>a[x-1] & a[x]<>0, \ print1(a[x]","); c++; ) ); return(c) }

Extensions

Definition corrected by Jonathan Sondow, Apr 21 2005