A275797 - OEIS

%I #55 Sep 17 2020 06:18:04

%S 11,19,41,409,419,449,499,811,1061,1069,1861,2459,2851,3061,3469,4091,

%T 4099,6449,6841,8011,8419,10211,11261,12251,12659,13669,14699,16649,

%U 18211,20809,20849,20899,22859,23869,26849,38611,42451,44491,46441,52259,53269,56249

%N Primes for which the concatenation of the digits in the even positions and the concatenation of the digits in the odd positions are squares.

%H Michael De Vlieger, <a href="/A275797/b275797.txt">Table of n, a(n) for n = 1..1000</a>

%H Jeremiah Bartz, Bruce Dearden, and Joel Iiams, <a href="https://arxiv.org/abs/1810.07895">Classes of Gap Balancing Numbers</a>, arXiv:1810.07895 [math.NT], 2018.

%H Jeremiah Bartz, Bruce Dearden, and Joel Iiams, <a href="https://ajc.maths.uq.edu.au/pdf/77/ajc_v77_p318.pdf">Counting families of generalized balancing numbers</a>, The Australasian Journal of Combinatorics (2020) Vol. 77, Part 3, 318-325.

%e 419 is prime and numbers 49 and 1 are square.

%e 12659 is prime and numbers 169 and 25 are square.

%t halfQ[n_, k_] := IntegerQ[Sqrt[FromDigits[IntegerDigits[n][[k ;; -1 ;; 2]]]]];

%t Select[Range[200000], PrimeQ[#] && halfQ[#, 1] && halfQ[#, 2] &] (* _Amiram Eldar_, Nov 05 2018 *)

%o (Perl) use ntheory ':all'; forprimes { my @d = split(//, $_); if (is_square(join('', map { $d[2*$_] } (0..$#d/2))) && is_square(join('', map { $d[2*$_+1] } (0..@d/2-1)))) { print "$_, " } } 10**6; # _Daniel Suteu_, Dec 03 2018

%o (PARI) isok(p) = {if (isprime(p), my (d=digits(p)); if (#d > 1, if (#d % 2, lo = #d\2+1; le = #d\2, le = #d\2; lo = #d\2); issquare(fromdigits(vector(le, k, d[2*k]))) && issquare(fromdigits(vector(lo, k, d[2*k-1])));););} \\ _Michel Marcus_, Dec 05 2018

%Y Cf. A000040, A000290.

%K nonn,base

%O 1,1

%A _Marius A. Burtea_, Oct 26 2018