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A297962 Take a prime, convert it to base 2. Remove its most significant digit and its least significant digit; repeat this process. a(n) is the least prime that, in the first n steps of this process, generates n primes.

%I A297962 #41 May 30 2022 16:33:15
%S A297962 13,59,631,7039,64063,761087,3619327,74347519,1577707519,22200700927,
%T A297962 1668463173631,290703062134783,3413184213843967,121597545150218239
%N A297962 Take a prime, convert it to base 2. Remove its most significant digit and its least significant digit; repeat this process. a(n) is the least prime that, in the first n steps of this process, generates n primes.
%C A297962 From _Jon E. Schoenfield_, Jan 11 2018: (Start)
%C A297962 a(15) <= 6937869050647642111;
%C A297962 a(16) <= 7088202090368908328959;
%C A297962 a(17) <= 348624306087955627410980863. (End)
%F A297962 Let C(p) be the result of removing the MSD and the LSD of a prime p. C(p) = (p - 2^floor(log_2(p)) - 1)/2.
%e A297962 a(1) = 13, because 13 in base 2 is 1101, 10 is the prime 2; and 13 is the least prime with this property.
%e A297962 a(2) = 59, because 59 = 111011_2, 1101_2 is the prime 13, 10_2 is the prime 2; and 59 is the least prime with this property.
%e A297962 a(3) = 631, because 631 = 1001110111_2, 111011_2 is the prime 59, 1101_2 is the prime 13, 10_2 is the prime 2; and 631 is the least prime with this property.
%p A297962 with(numtheory): P:=proc(q) local a,i,k,n,ok,x; x:=1;
%p A297962 for n from 1 to q do for k from x to q do
%p A297962 a:=convert(ithprime(k),binary,decimal);
%p A297962 ok:=1; for i from 1 to n do a:=trunc(a/10) mod 10^(ilog10(a)-1);
%p A297962 if not isprime(convert(a,decimal,binary)) then ok:=0; break; fi; od;
%p A297962 if ok=1 then x:=k; print(ithprime(k)); break; fi; od; od; end: P(10^10);
%t A297962 With[{s = Map[LengthWhile[#, PrimeQ] &@ NestWhileList[((# - 2^Floor@ Log2@ #) - 1)/2 &, #, # > 2 &] &, Prime@ Range[2^18]]}, Map[Prime@ First@ FirstPosition[s, #] &, Range@ Max@ s]] (* _Michael De Vlieger_, Jan 10 2018 *)
%o A297962 (Java) private static BigInteger SearchAn() { BigInteger BIW, BICore; int tN=10; int j3; static final BigInteger BILim = new BigInteger("1000000000"); static final BigInteger BI2 = new BigInteger("2");for (BIW=BI2; BIW.compareTo(BILim)<0; BIW=BIW.add(BI2)) { BICore = BIW;  for (j3=1;j3<tN+1;j3++) { if (!(BICore.isProbablePrime(10))) break; else { BICore=((BICore.subtract(BI2.pow(BICore.bitLength()-1)).subtract(BI1))).divide(BI2); } } if (j3==tN+1) { return BIW; } return BigInteger.ZERO; } }
%Y A297962 Cf. A296806, A296807.
%K A297962 nonn,base,hard,more
%O A297962 1,1
%A A297962 _Paolo Iachia_, _Paolo P. Lava_, Jan 09 2018
%E A297962 a(10)-a(14) from _Jon E. Schoenfield_, Jan 11 2018