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%I #3 Mar 05 2022 13:09:09
%S 29,76,2289,3796,6369,8756,16736,19696,24900,28484,77529,83761,94169,
%T 222889,887556,22228889,88875556,112594641,368762025,651177616
%N Numbers n with property that concatenation (2*n+1)//n is a square.
%C If n d-digit number, (2*n+1) * 10^d + n = n * (2 * 10^d + 1) + 10^d = N^2
%C Sequence is infinite, two infinite "families" of such numbers n are:
%C (a) n = 8_(k)75_(k)6, 2 * n + 1 = 17_(k)51_(k)3, N = 2 * 6_(k+1)16_(k-1)7,
%C (b) n = 2_(k+1)8_(k)9, 2 * n + 1 = 4_(k)57_(k)9, N = 6_(k)76_(k)7, (k = 1, 2, ...)
%C List of (2*n+1)//n = N^2:
%C 59//29 = 7^2 x 11^2, 153//76 = 2^4 x 31^2, 4579//2289 = 67^2 x 101^2,
%C 7593//3796 = 2^2 x 4357^2, 12739//6369 = 11287^2, 17513//8756 = 2^2 x 13^2 x 509^2,
%C 33473//16736 = 2^18 x 113^2, 39393//19696 = 2^4 x 13^2 x 17^2 x 71^2, 49801//24900,
%C 56969//28484 = 2^2 x 13^2 x 2903^2, 155059//77529 = 7^2 x 17789^2, 167523//83761 = 347^2 x 373^2,
%C 188339//94169 = 19^2 x 31^2 x 233^2, 445779//222889 = 7^2 x 11^2 x 13^2 x 23^2 x 29^2,
%C 1775113//887556 = 2^2 x 666167^2, 44457779//22228889 = 59^2 x 73^2 x 113^2 x 137^2,
%C 177751113//88875556 = 2^2 x 66661667 ^ 2, 225189283//112594641 = 23^2 x 83^2 x 331^2 x 751^2,
%C 737524051//368762025 = 5^2 x 2161^2 x 79481^2, 1302355233//651177616 = 2^4 x 285301949^2
%D J. Buchmann, U. Vollmer: Binary Quadratic Forms, Springer, Berlin, 2007
%D L. E. Dickson: History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications, 2005
%e n = 29 = prime(10) is 1st term: 2 * n + 1 = 59 = prime(17), 5929 = 77^2, 59//29 a concatenation of two primes
%e n = 6369 (composite) is 4th term: 2 * n + 1 = 12739 = prime(1520), N = 11287 = prime(1365), only prime base of above terms
%t Select[Range[6512*10^5],IntegerQ[Sqrt[(2 #+1)10^IntegerLength[#]+#]]&] (* _Harvey P. Dale_, Mar 05 2022 *)
%Y A174370, A174454, A174926, A176130
%K base,nonn,uned
%O 1,1
%A Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 11 2010
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