A321151 Primes that yield squares after deletion of their zero digits.

409, 1021, 1069, 1201, 1609, 2089, 3061, 5209, 9601, 10069, 10369, 18049, 20089, 20809, 37021, 37201, 40009, 44089, 44809, 50329, 50929, 52009, 59029, 59209, 60889, 62401, 70921, 79201, 96001, 100069, 100609, 101449, 102001, 102769, 103069, 104161, 106129, 106801, 108769, 109321

Offset: 1

Author

Marius A. Burtea, Nov 23 2018

Keywords

• nonn
• base

Comments

Subsequence of A056709. The squares divisible by 2, 3 or 5 cannot be obtained. Most of the squares obtained seem to be squares of prime or semiprime numbers. Among the first 399 squares obtained are 289 squares of prime numbers, 104 squares of semiprimes and 6 other squares; these squares were obtained by testing the prime numbers up to 10^7. Deletion of the zero digits from primes up to 10^8 yields 929 squares of prime numbers, 506 squares of semiprimes and 47 other squares. Do similar results occur when larger primes are considered?

From David A. Corneth, Nov 26 2018: (Start)

Terms can be obtained by listing squares coprime to 30, inserting zeros between digits, and testing the primality of the resulting numbers.

Records for omega(s) where s is a square producing a term occur at terms 409, 50929, 10713481, 3601722361, 1531869148081, 807916258118689. (End)

Examples

			      409 is prime and   49 =  7^2 is a square.
     9601 is prime and  961 = 31^2 is a square.
    20809 is prime and  289 = 17^2 is a square.
    10069 is prime and  169 = 13^2 is a square.
   103069 is prime and 1369 = 37^2 is a square.
  1030069 is prime and 1369 = 37^2 is a square.
		

Links

• David A. Corneth, Table of n, a(n) for n = 1..17124 (terms < 10^10)
• David A. Corneth, PARI program

Crossrefs

Cf. A000040, A000290, A052041, A056709, A256186.

Programs

Mathematica

aQ[n_] := PrimeQ[n] && IntegerQ[Sqrt[FromDigits[Select[IntegerDigits[n], #!=0  &]]]]; Select[Range[100000], aQ] (* Amiram Eldar, Nov 25 2018 *)

PARI

isok(p) = isprime(p) && issquare(fromdigits(select(x->x, digits(p)))); \\ Michel Marcus, Nov 26 2018

PARI

\\ See Corneth link \\ David A. Corneth, Nov 26 2018