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

1,1

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

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

Formula

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

Example

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.

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

Prog

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

Crossrefs

Cf. A167416, A167417.

Supersequence of A345314.

Sequence in context: A061246 A353102 A068493 * A184328 A260271 A275797

Adjacent sequences: A167532 A167533 A167534 * A167536 A167537 A167538

Keyword

nonn,base

Author

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

Extensions

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

Status

approved