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

5, 17, 53, 151, 157, 163, 173, 463, 467, 491, 523, 541, 1451, 1471, 1483, 1489, 1499, 1511, 1549, 1553, 1567, 1609, 1657, 1721, 1723, 4583, 4597, 4621, 4649, 4663, 4673, 4703, 4729, 4751, 4759, 4787, 4813, 4937, 4943, 4973, 4993, 5039, 5077, 5227, 5297

Offset: 1

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 29 2009

Keywords

• nonn
• base

Comments

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

Examples

			i) The prime 47 has the square 47^2=2209, but 209=11*19 is not prime, so 47 is not in the sequence.
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.
iii) The square of the prime 53 is 2809, i.e., 2 followed by the prime 809, so 53 is in the sequence.
		

References

• Wladyslaw Narkiewicz, The development of prime number theory. From Euclid to Hardy and Littlewood. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000.
• I. Niven, H. S. Zuckerman and H. L. Montgomery, An Introduction to the Theory of Numbers (5th ed.). Wiley Text Books, 1991.
• Paulo Ribenboim, The New Book of Prime Number Records. Springer, 1996.

Links

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

Crossrefs

Cf. A001248.

Programs

Maple

count:= 0: N:= 100: Res:= NULL:
for d from 1 while count < N do
  p:= floor(sqrt(2*10^d+10^(d-1)));
  while count < N do
    p:= nextprime(p);
    if p^2 >= 3*10^d then break fi;
    q:= p^2 - 2*10^d;
    if isprime(q) then
       count:= count+1;
       Res:= Res, p;
    fi
od od:
Res; # Robert Israel, Mar 06 2018

Mathematica

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 *)

Python

from sympy import isprime, primerange
def ok(p):
  s = str(p*p); return s[0] == '2' and s[1] != '0' and isprime(int(s[1:]))
print(list(filter(ok, primerange(2, 5298)))) # Michael S. Branicky, May 17 2021

Extensions

1451 inserted, and sequence extended beyond 4673, by R. J. Mathar, Apr 01 2009

Edited by R. J. Mathar, Apr 06 2009