A167535 Primes which are the concatenation of two squares (in decimal notation).

11, 19, 41, 149, 181, 251, 449, 491, 499, 641, 811, 1009, 1289, 1361, 1699, 2251, 2549, 4001, 4289, 4441, 4729, 6449, 6481, 6761, 7841, 8419, 9001, 9619, 10891, 11369, 11681, 12149, 12251, 12401, 12601, 12809, 13249, 13691, 13721, 14449, 14489

Offset: 1

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 06 2009

Keywords

• nonn
• base

Comments

Necessarily a(n) has to end with 1 or 9.

It is not known if the sequence is infinite.

The Bunyakovsky conjecture implies that for every b coprime to 10, there are infinitely many terms where the second square is b^2. - Robert Israel, Jun 17 2021

Intersection of A191933 and A000040; A193095(a(n)) > 0 and A010051(a(n))=1. - Reinhard Zumkeller, Jul 17 2011

Examples

			11 = 1^2 * 10 + 1^2, 149 = 1^2 * 10^2 + 7^2, 1361 = 1^2 * 10^3 + 19^2.
14401 = 12^2 * 10^2 + 1^2 is not a term because included "0" (1^2=1 is 1-digit).
14449 = 12^2 * 10^2 + 7^2 = 38^2 * 10 + 3^2 is the smallest prime with 2 such representations.
		

References

• Richard E. Crandall, Carl Pomerance, Prime Numbers, Springer 2005.
• Wladyslaw Narkiewicz, The Development of Prime Number Theory from Euclid to Hardy and Littlewood, Springer 2000.
• Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A167416, A167417.

Supersequence of A345314.

Programs

Haskell

a167535 n = a167535_list !! (n-1)
a167535_list = filter ((> 0) . a193095) a000040_list
-- Reinhard Zumkeller, Jul 17 2011

Maple

zcat:= proc(a,b) 10^(1+ilog10(b))*a+b end proc;
S:= select(t -> t <= 10^7 and isprime(t), {seq(seq(zcat(a^2,b^2),a=1..10^3),b=1..10^3,2)}):
sort(convert(S,list)); # Robert Israel, Jun 17 2021

PARI

is_A167535(n)={ my(t=1); isprime(n) && while(n>t*=10, apply(issquare,divrem(n,t))==[1,1]~ && n%t*10>=t && return(1))}
forprime(p=1,default(primelimit), is_A167535(p) && print1(p",")) \\ M. F. Hasler, Jul 24 2011

Python

from sympy import isprime
def aupto(lim):
    s = list(i**2 for i in range(1, int(lim**(1/2))+2))
    t = set(int(str(a)+str(b)) for a in s for b in s)
    return sorted(filter(isprime, filter(lambda x: x<=lim, t)))
print(aupto(15000)) # Michael S. Branicky, Jun 17 2021

Formula

a(n) = m^2 * 10^k + n^2 for a k-digit square number n^2.

Extensions

11369 inserted by R. J. Mathar, Nov 07 2009