A100718 Composite numbers C(p) such that p and C(p)-p are primes.

8, 10, 14, 30, 54, 58, 62, 66, 82, 108, 114, 120, 178, 182, 204, 210, 318, 324, 330, 352, 366, 430, 506, 544, 560, 586, 596, 616, 704, 738, 792, 858, 870, 914, 918, 960, 988, 990, 1026, 1030, 1062, 1164, 1170, 1194, 1404, 1442, 1446, 1462, 1464, 1470, 1498

Offset: 1

Author

Robin Garcia, Dec 11 2004

Keywords

• nonn

Comments

Nextprime(C(n)) = P(C(n) - n) = (C(n) - n)-th prime.

A proof that the sequence is infinite would be nice.

Examples

			a(4)=30 because C(19)=30=19+11, 19 and 11 are prime and P(11)=31=nextprime(30).
		

Links

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

Programs

Maple

R:= NULL: count:= 0: m:= 0:
for c from 2 while count < 100 do
  if isprime(c) then next fi;
  m:= m+1;
  if isprime(m) and isprime(c-m) then
    count:= count+1; R:= R,c
  fi
od:
R; # Robert Israel, Nov 23 2024

Mathematica

Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n]; Composite /@ Select[ Prime[ Range[ 205]], PrimeQ[ Composite[ # ] - # ] &] (* Robert G. Wilson v, Dec 11 2004 *)
Reap[For[n = 4; p = 1, n <= 1500, n++, If[! PrimeQ[n], If[PrimeQ[p] && PrimeQ[n-p], Sow[n]]; p++]]] [[2, 1]]  (* Jean-François Alcover, Jul 18 2013 *)

Formula

C(n)=n+k where k is such that nextprime(C(n))=k-th prime.

Extensions

Edited and extended by Robert G. Wilson v, Dec 11 2004