A167416 -id:A167416 - cp's OEIS frontend

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

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

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

			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.

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.

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

Cf. A167416, A167417.

Supersequence of A345314.

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

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

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

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

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

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

A167417 Largest prime concatenation of the first n primes, or 0 if no such prime exists.

Original entry on oeis.org

2, 23, 523, 7523, 751123, 75311213, 7523171311, 753217131911, 75323219131117, 0, 753312923219111713, 75373312923192171311, 7541373132923217111319, 754341373132923192171311, 75474341373132923211171319

Offset: 1

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 03 2009

a(10) doesn't exist, because the sum of digits of the first 10 primes (2+3+5+7+(1+1)+(1+3)+(1+7)+(1+9)+(2+3)+(2+9)) = 57 is a multiple of 3.

			The only prime concatenations of the first n primes for n = 1..3 are a(1)=2, a(2)=23, and a(3)=523.
For n=4, the only prime concatenations of 2, 3, 5, and 7 are 2357, 2753, 3257, 3527, 5237, 5273, 7253, 7523; the largest of these is a(4) = 7523.

Richard E. Crandall and Carl Pomerance, Prime Numbers, Springer 2005.
Paulo Ribenboim, The New Book of Prime Number Records, Springer 1996.
A. Weil, Number theory: an approach through history, Birkhäuser 1984.

Gleb Ivanov, Python program for large terms.

Cf. A175429, A177275, A134966, A167416.

from sympy import sieve, isprime
from itertools import permutations
for n in range(1, 14):
    sieve.extend_to_no(n)
    p = list(map(str, list(sieve._list)))[:n]
    mint = 0
    for i in permutations(p, len(p)):
        t = int(''.join(i))
        if  t > mint and isprime(t):
            mint = t
    print(mint, end = ', ') # Gleb Ivanov, Dec 05 2021

Edited by Charles R Greathouse IV, Apr 28 2010

Several terms corrected and a(11)-a(15) from Gleb Ivanov, Dec 05 2021

cp's OEIS Frontend

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

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