A084006 Squares arising as a concatenation of k and 9's complement of k.

36, 81, 1089, 4356, 9801, 110889, 443556, 998001, 11108889, 44435556, 99980001, 1111088889, 4444355556, 9999800001, 111110888889, 444443555556, 999998000001, 11111108888889, 44444435555556, 99999980000001

Offset: 1

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 23 2003

Keywords

• base
• nonn

Comments

From Robert Israel, Sep 09 2020: (Start)

Numbers of the form j^2*x*(10^k-1) where x = A007913(10^k-1) and 10^(k-1)+1 <= j^2*x <= 10^k-1.

If k >= 2 is not in A046412, there are only three terms with 2*k digits, namely (10^k-1)^2/9, 4*(10^k-1)^2/9, and 9*(10^k-1)^2/9.

The first term not of one of those three forms is a(25)=197530863802469136.

(End)

Examples

			1089 = 33^2 is a concatenation of 10 and 89, 10+89 = 99.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..367

Crossrefs

Cf. A007913, A046412, A084004, A084005, A084007.

Programs

Maple

f:= proc(k) local F,x,p,t;
  p:= 10^k-1;
  F:= select(t -> t[2]::odd, ifactors(p)[2]);
  x:= mul(t[1],t=F);
  seq(j^2*x*p, j=ceil(sqrt((10^(k-1)+1)/x))..floor(sqrt(p/x)))
end proc:
map(f, [$1..20]); # Robert Israel, Sep 09 2020

Python

from itertools import count, islice
from math import prod, isqrt
from sympy import factorint
def A084006_gen(): # generator of terms
    for l in count(1):
        m = 10**l-1
        x = prod(p for p, e in factorint(m).items() if e&1)
        yield from (j**2*x*m for j in range(isqrt(10**(l-1)//x)+1,isqrt(m//x)+1))
A084006_list = list(islice(A084006_gen(),20)) # Chai Wah Wu, Mar 20 2025

Extensions

More terms from Ray Chandler, May 31 2003

Offset changed by Robert Israel, Sep 09 2020