A176185 Numbers n with property that concatenation (2*n+1)//n is a square.

29, 76, 2289, 3796, 6369, 8756, 16736, 19696, 24900, 28484, 77529, 83761, 94169, 222889, 887556, 22228889, 88875556, 112594641, 368762025, 651177616

Offset

1,1

Comments

If n d-digit number, (2*n+1) * 10^d + n = n * (2 * 10^d + 1) + 10^d = N^2

Sequence is infinite, two infinite "families" of such numbers n are:

(a) n = 8_(k)75_(k)6, 2 * n + 1 = 17_(k)51_(k)3, N = 2 * 6_(k+1)16_(k-1)7,

(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, ...)

List of (2*n+1)//n = N^2:

59//29 = 7^2 x 11^2, 153//76 = 2^4 x 31^2, 4579//2289 = 67^2 x 101^2,

7593//3796 = 2^2 x 4357^2, 12739//6369 = 11287^2, 17513//8756 = 2^2 x 13^2 x 509^2,

33473//16736 = 2^18 x 113^2, 39393//19696 = 2^4 x 13^2 x 17^2 x 71^2, 49801//24900,

56969//28484 = 2^2 x 13^2 x 2903^2, 155059//77529 = 7^2 x 17789^2, 167523//83761 = 347^2 x 373^2,

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,

1775113//887556 = 2^2 x 666167^2, 44457779//22228889 = 59^2 x 73^2 x 113^2 x 137^2,

177751113//88875556 = 2^2 x 66661667 ^ 2, 225189283//112594641 = 23^2 x 83^2 x 331^2 x 751^2,

737524051//368762025 = 5^2 x 2161^2 x 79481^2, 1302355233//651177616 = 2^4 x 285301949^2

References

J. Buchmann, U. Vollmer: Binary Quadratic Forms, Springer, Berlin, 2007

L. E. Dickson: History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications, 2005

Links

Table of n, a(n) for n=1..20.

Example

n = 29 = prime(10) is 1st term: 2 * n + 1 = 59 = prime(17), 5929 = 77^2, 59//29 a concatenation of two primes
n = 6369 (composite) is 4th term: 2 * n + 1 = 12739 = prime(1520), N = 11287 = prime(1365), only prime base of above terms

Mathematica

Select[Range[6512*10^5], IntegerQ[Sqrt[(2 #+1)10^IntegerLength[#]+#]]&] (* Harvey P. Dale, Mar 05 2022 *)

Crossrefs

A174370, A174454, A174926, A176130

Sequence in context: A071110 A263534 A273360 * A044167 A044548 A144584

Adjacent sequences: A176182 A176183 A176184 * A176186 A176187 A176188

Keyword

base,nonn,uned

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Apr 11 2010

Status

approved