A151991 Numbers k with the property that (k-x)*(k-y)*(k-z) = x*y*z has no integer solutions 0 < x,y,z < k.

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 37, 39, 41, 43, 47, 49, 51, 53, 57, 59, 61, 67, 69, 71, 73, 79, 81, 83, 87, 89, 93, 95, 97, 101, 103, 107, 109, 111, 113, 115, 121, 123, 125, 127, 129, 131, 133, 137, 139, 141, 145, 147, 149, 151, 155, 157, 159

Offset: 1

Author

Nurdin Takenov (greanvert(AT)gmail.com), Sep 14 2009

Keywords

• nonn

Comments

All odd primes are terms of this sequence.

Examples

			15 is not a term of this sequence because (15-x)*(15-y)*(15-z) = x*y*z has the solution (5,5,12).
		

Crossrefs

Cf. A065091 (odd primes).

Programs

PARI

is(n)=for(x=1,n-1,for(y=1,x, my(t=(n-x)*(n-y),z=t*n/(x*y+t)); if(denominator(z)==1 && 0 < z && z < n, return(0)))); 1 \\ Charles R Greathouse IV, Dec 09 2014

Python

def exis(n):
    for x in range(1,n):
        for y in range(x+1):
            for z in range(y+1):
                if x*y*z==(k-x)*(k-y)*(k-z):
                    return True
    return False
for k in range(1, 200, 2):
    if not exis(k):
        print(str(k), end=',')

Extensions

Edited by Charles R Greathouse IV, Dec 09 2014