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%I #8 Jan 21 2019 19:01:55
%S 3,5,11,17,31,41,71,79,101,127,139,149,179,199,211,229,241,269,277,
%T 307,359,383,431,449,541,773,829,881,1259,1307,1327,1493,1831,1933,
%U 2141,2551,3373,3947,4127,4831,4957,5021,5153,5323,5431,5569,5813,6091,6329
%N Primes p such that for some k the number of terms > 0 and < 1 in the Farey sequence of order k is p.
%C Or, prime numbers of the form Sum(phi(j), j=2..n). - _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.
%C Does this sequence have an infinite number of terms?
%e 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 member.
%o (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) }
%Y Cf. A078334, A005728, A101300, A000010.
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, Aug 16 2003
%E Definition corrected by _Jonathan Sondow_, Apr 21 2005
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