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A158896 Primes whose squares are a concatenation of 2 with some prime.

%I A158896 #19 May 18 2021 01:23:21
%S A158896 5,17,53,151,157,163,173,463,467,491,523,541,1451,1471,1483,1489,1499,
%T A158896 1511,1549,1553,1567,1609,1657,1721,1723,4583,4597,4621,4649,4663,
%U A158896 4673,4703,4729,4751,4759,4787,4813,4937,4943,4973,4993,5039,5077,5227,5297
%N A158896 Primes whose squares are a concatenation of 2 with some prime.
%C A158896 Leading zeros of the prime within the concatenation are not admitted. Example: 449^2=201601 is a square of a prime which is 2 concatenated with a zero-padded prime 1601. 449 is not in the sequence. - _R. J. Mathar_, Apr 06 2009
%D A158896 Wladyslaw Narkiewicz, The development of prime number theory. From Euclid to Hardy and Littlewood. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000.
%D A158896 I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers (5th ed.). Wiley Text Books, 1991.
%D A158896 Paulo Ribenboim, The New Book of Prime Number Records. Springer, 1996.
%H A158896 Robert Israel, <a href="/A158896/b158896.txt">Table of n, a(n) for n = 1..10000</a>
%e A158896 i) The prime 47 has the square 47^2=2209, but 209=11*19 is not prime, so 47 is not in the sequence.
%e A158896 ii) If we attach 2 to the prime p=401 we get 2401=49^2, but 49=7^2 is not a prime, so there is no contribution to the sequence.
%e A158896 iii) The square of the prime 53 is 2809, i.e., 2 followed by the prime 809, so 53 is in the sequence.
%p A158896 count:= 0: N:= 100: Res:= NULL:
%p A158896 for d from 1 while count < N do
%p A158896   p:= floor(sqrt(2*10^d+10^(d-1)));
%p A158896   while count < N do
%p A158896     p:= nextprime(p);
%p A158896     if p^2 >= 3*10^d then break fi;
%p A158896     q:= p^2 - 2*10^d;
%p A158896     if isprime(q) then
%p A158896        count:= count+1;
%p A158896        Res:= Res, p;
%p A158896     fi
%p A158896 od od:
%p A158896 Res; # _Robert Israel_, Mar 06 2018
%t A158896 okQ[n_]:=Module[{idn=IntegerDigits[n^2]},First[idn]==2&& idn[[2]]!=0 && PrimeQ[FromDigits[Rest[idn]]]]; Select[Prime[Range[750]],okQ] (* _Harvey P. Dale_, Jul 22 2011 *)
%o A158896 (Python)
%o A158896 from sympy import isprime, primerange
%o A158896 def ok(p):
%o A158896   s = str(p*p); return s[0] == '2' and s[1] != '0' and isprime(int(s[1:]))
%o A158896 print(list(filter(ok, primerange(2, 5298)))) # _Michael S. Branicky_, May 17 2021
%Y A158896 Cf. A001248.
%K A158896 nonn,base
%O A158896 1,1
%A A158896 Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 29 2009
%E A158896 1451 inserted, and sequence extended beyond 4673, by _R. J. Mathar_, Apr 01 2009
%E A158896 Edited by _R. J. Mathar_, Apr 06 2009