A167535 - OEIS

%I #23 Jun 18 2021 01:14:23

%S 11,19,41,149,181,251,449,491,499,641,811,1009,1289,1361,1699,2251,

%T 2549,4001,4289,4441,4729,6449,6481,6761,7841,8419,9001,9619,10891,

%U 11369,11681,12149,12251,12401,12601,12809,13249,13691,13721,14449,14489

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

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

%C It is not known if the sequence is infinite.

%C 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

%C Intersection of A191933 and A000040; A193095(a(n)) > 0 and A010051(a(n))=1. - _Reinhard Zumkeller_, Jul 17 2011

%D Richard E. Crandall, Carl Pomerance, Prime Numbers, Springer 2005.

%D Wladyslaw Narkiewicz, The Development of Prime Number Theory from Euclid to Hardy and Littlewood, Springer 2000.

%D Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996.

%H Reinhard Zumkeller, <a href="/A167535/b167535.txt">Table of n, a(n) for n = 1..500</a>

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

%e 11 = 1^2 * 10 + 1^2, 149 = 1^2 * 10^2 + 7^2, 1361 = 1^2 * 10^3 + 19^2.

%e 14401 = 12^2 * 10^2 + 1^2 is not a term because included "0" (1^2=1 is 1-digit).

%e 14449 = 12^2 * 10^2 + 7^2 = 38^2 * 10 + 3^2 is the smallest prime with 2 such representations.

%p zcat:= proc(a,b) 10^(1+ilog10(b))*a+b end proc;

%p 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)}):

%p sort(convert(S,list)); # _Robert Israel_, Jun 17 2021

%o (Haskell)

%o a167535 n = a167535_list !! (n-1)

%o a167535_list = filter ((> 0) . a193095) a000040_list

%o -- _Reinhard Zumkeller_, Jul 17 2011

%o (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))}

%o forprime(p=1,default(primelimit), is_A167535(p) && print1(p",")) \\ _M. F. Hasler_, Jul 24 2011

%o (Python)

%o from sympy import isprime

%o def aupto(lim):

%o     s = list(i**2 for i in range(1, int(lim**(1/2))+2))

%o     t = set(int(str(a)+str(b)) for a in s for b in s)

%o     return sorted(filter(isprime, filter(lambda x: x<=lim, t)))

%o print(aupto(15000)) # _Michael S. Branicky_, Jun 17 2021

%Y Cf. A167416, A167417.

%Y Supersequence of A345314.

%K nonn,base

%O 1,1

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

%E 11369 inserted by _R. J. Mathar_, Nov 07 2009