%I #26 Aug 22 2024 02:42:28

%S 1,0,0,1,0,4,2,9,1,3,8,25,4,36,18,1,2,64,6,81,16,4,50,121,1,20,72,9,

%T 36,196,2,225,4,11,128,1,12,324,162,13,5,400,8,441,100,3,242,529,1,63,

%U 40,17,144,676,18,9,7,19,392,841,4,900,450,1,8,16,22,1089,256,23,2,1225

%N Least number k such that k (k + n) is a perfect square, or 0 if impossible.

%C Impossible only for 1, 2 and 4. k equals 1 when n is in A005563. k equals 2 when n is in A054000.

%C Let k*(k+n)= c*c, gcd(n,k,c)=1 . Then primitive triples (n,k,c) are of the form : 1) n is prime. (n,k,c)=( p, (p*p-2*p+1)/4, (p*p-1)/4 ) 2) n=(c/t)*(c/t)- t*t, n is not a prime, t positive integer. (n,k,c)=( (c/t)*(c/t)- t*t, t*t, c ). [_Ctibor O. Zizka_, May 04 2009]

%H Chai Wah Wu, <a href="/A067721/b067721.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..2000 from Carmine Suriano)

%e a(7) = 9 because 9 (7+9) = 144 = 12^2.

%t Do[k = 1; While[ !IntegerQ[ Sqrt[ k (k + n)]], k++ ]; Print[k], {n, 5, 75} ]

%o (Python)

%o from itertools import takewhile

%o from collections import deque

%o from sympy import divisors

%o def A067721(n): return ((a:=next(iter(deque((d for d in takewhile(lambda d:d<n, divisors(n**2)) if not (d-n**2//d)&3),1)),0))+(n**2//a-(n<<1) if a else 0)>>2) if n else 1 # _Chai Wah Wu_, Aug 21 2024

%Y Cf. A005563, A007913, A054000, A067632, A076942.

%K nonn

%O 0,6

%A _Robert G. Wilson v_, Feb 05 2002