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Composite numbers C(p) such that p and C(p)-p are primes.
2

%I #14 Nov 24 2024 03:38:26

%S 8,10,14,30,54,58,62,66,82,108,114,120,178,182,204,210,318,324,330,

%T 352,366,430,506,544,560,586,596,616,704,738,792,858,870,914,918,960,

%U 988,990,1026,1030,1062,1164,1170,1194,1404,1442,1446,1462,1464,1470,1498

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

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

%C A proof that the sequence is infinite would be nice.

%H Robert Israel, <a href="/A100718/b100718.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%p R:= NULL: count:= 0: m:= 0:

%p for c from 2 while count < 100 do

%p if isprime(c) then next fi;

%p m:= m+1;

%p if isprime(m) and isprime(c-m) then

%p count:= count+1; R:= R,c

%p fi

%p od:

%p R; # _Robert Israel_, Nov 23 2024

%t Composite[n_Integer] := FixedPoint[n + PrimePi[ # ] + 1 &, n]; Composite /@ Select[ Prime[ Range[ 205]], PrimeQ[ Composite[ # ] - # ] &] (* _Robert G. Wilson v_, Dec 11 2004 *)

%t 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 *)

%K nonn

%O 1,1

%A _Robin Garcia_, Dec 11 2004

%E Edited and extended by _Robert G. Wilson v_, Dec 11 2004