A255769 Primes p such that there are a prime number of composite numbers less than p.

7, 11, 23, 31, 47, 59, 67, 83, 97, 109, 137, 149, 167, 179, 197, 211, 233, 269, 331, 347, 353, 367, 389, 419, 431, 439, 587, 617, 739, 751, 829, 859, 907, 919, 977, 991, 1009, 1031, 1039, 1063, 1117, 1171, 1187, 1237, 1319, 1327, 1427, 1447, 1471, 1499, 1553, 1567, 1723, 1901, 1913, 1933, 2207, 2221, 2269, 2293, 2333

Offset: 1

Author

Antonio Gimenez, Jul 11 2015

Keywords

• nonn

Examples

			There are two composite numbers less than 7, namely, 4 and 6, and 2 is prime. Therefore 7 is a member of the sequence.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A072677.

Programs

Maple

c:= proc(n) option remember; `if`(n<4, 0,
      c(n-1)+`if`(isprime(n-1), 0, 1))
    end:
a:= proc(n) option remember; local p;
      p:= `if`(n=1, 1, a(n-1));
      do p:= nextprime(p);
         if isprime(c(p)) then break fi
      od; p
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 23 2015

Mathematica

fQ[n_]:=PrimeQ[n-PrimePi[n]-1];Select[Prime[Range@400],fQ[#]&] (* Ivan N. Ianakiev, Jul 12 2015 *)

PARI

is_ok(n)=my(i,k=0); for(i=2,n-1,if(bigomega(i)>1,k++)); isprime(k)&&isprime(n);
first(m)=my(i=1,v=vector(m),k=0);while(i<=m,if(is_ok(k), v[i]=k;i++);k++);v; \\ Anders Hellström, Jul 29 2015

PARI

listp(nn)=forprime(p=2, nn, if (isprime(p - primepi(p) - 1), print1(p, ", "));); \\ Michel Marcus, Aug 27 2016

PARI

list(lim)=my(v=List(),n=1); forprime(p=2,lim, if(isprime(p - n++), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Aug 28 2016

Extensions

a(16)-a(61) from Ivan N. Ianakiev, Jul 12 2015