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a(n) is the least prime that can be written in exactly n ways as s+2*p where s is a semiprime and p is a prime, or 0 if no such prime exists.
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%I #10 Oct 29 2021 02:21:12

%S 2,13,19,31,43,59,61,83,127,149,151,181,257,167,241,223,251,401,227,

%T 311,419,383,347,499,479,547,631,593,523,599,563,727,691,743,857,829,

%U 953,863,853,827,1093

%N a(n) is the least prime that can be written in exactly n ways as s+2*p where s is a semiprime and p is a prime, or 0 if no such prime exists.

%C a(41) > 10^5 if it is not 0.

%C Conjecture: First zeros of the sequence are a(41), a(666), a(1277), a(2701), ... - _Giorgos Kalogeropoulos_, Sep 23 2021

%e a(3) = 31 because the prime 31 can be obtained in exactly 3 ways:

%e 31 = 9+2*11 = 21+2*5 = 25+2*3

%e and this is the least such prime.

%p N:= 10000: # to search values up to N

%p P:= select(isprime, [2,seq(i,i=3..N,2)]):

%p S:= select(t -> numtheory:-bigomega(t)=2, [$1..N]):

%p V:= Vector(N):

%p for p in P do

%p for s in S do

%p r:= s+2*p;

%p if r > N then break fi;

%p V[r]:= V[r]+1

%p od

%p od:

%p m:= min({$0..max(V[P])} minus convert(V[P],set))-1:

%p M:= Array(0..m):

%p for p in P do

%p v:= V[p];

%p if v <= m and M[v] = 0 then M[v]:= p fi

%p od:

%p convert(M,list);

%t Table[k=2;While[Length@Select[k-2Prime@Range@PrimePi@Floor[k/2],PrimeOmega@#==2&]!=n,k=NextPrime@k];k,{n,0,40}] (* _Giorgos Kalogeropoulos_, Sep 23 2021 *)

%K nonn,more

%O 0,1

%A _J. M. Bergot_ and _Robert Israel_, Aug 24 2021