A067721 Least number k such that k (k + n) is a perfect square, or 0 if impossible.

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, 36, 196, 2, 225, 4, 11, 128, 1, 12, 324, 162, 13, 5, 400, 8, 441, 100, 3, 242, 529, 1, 63, 40, 17, 144, 676, 18, 9, 7, 19, 392, 841, 4, 900, 450, 1, 8, 16, 22, 1089, 256, 23, 2, 1225

Offset: 0

Author

Robert G. Wilson v, Feb 05 2002

Keywords

• nonn

Comments

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

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]

Examples

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

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Carmine Suriano)

Crossrefs

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

Programs

Mathematica

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

Python

from itertools import takewhile
from collections import deque
from sympy import divisors
def A067721(n): return ((a:=next(iter(deque((d for d in takewhile(lambda d:d>2) if n else 1 # Chai Wah Wu, Aug 21 2024