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
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