A375323 Natural numbers k for which there exist distinct nonzero naturals a,b,c, such that k = a + b + c and (a + b)*(b + c)*(c + a) is a perfect cube.

10, 19, 20, 30, 37, 38, 40, 46, 47, 50, 57, 60, 61, 66, 67, 68, 70, 74, 75, 76, 80, 90, 91, 92, 94, 95, 100, 101, 107, 109, 110, 111, 113, 114, 120, 122, 127, 129, 130, 131, 132, 133, 134, 136, 138, 139, 140, 141, 148, 150, 152, 160, 167, 169, 170, 171, 180, 182

Offset: 1

Author

Marius A. Burtea, Sep 15 2024

Keywords

• nonn

Comments

The sequence is inspired by problem 1, Junior Balkan Team Selection Tests - Romania 2023, Brasov, 13.04.2023, (see link).

If k >= 1 is a term, then for any m >= 1 the number m*k is also a term.

Examples

			10 = 1 + 2 + 7 and (1 + 2)*(2 + 7)*(7 + 1) = 27*8 = 6^3, is a cube, so 10 is a term.
19 = 1 + 7 + 11 and (1 + 7)*(7 + 11)*(11 + 1) = 8*18*12 = 2^3*6^3 = 12^3, is a cube, so 19 is a term.
57 = 3 + 7 + 47 and (3 + 7)*(7 + 47)*(47 + 3) = 10*54*50 = 27*1000 = 30^3, is a cube. Also 57 = 3 + 21 + 33 and (3 + 21)*(21 + 33)*(33 + 3) = 24*54*36 = 36^2*36 = 36^3, is a cube.
		

Links

• Junior Balkan Team Selection Tests - Romania 2023, Problems

Programs

Magma

[n:n in [1..200]|exists(u){:a in [1..n-2],b in [1..n-2]|a lt b and #{a,b,n-a-b} eq 3 and n-a-b gt 0 and IsPower((n-a)*(n-b)*(a+b),3)}];

Python

from itertools import count, islice
from sympy import integer_nthroot
def A375323_gen(startvalue=1): # generator of terms >= startvalue
    return (k for k in count(max(startvalue,1)) if any(integer_nthroot(a*(a*(m:=b-k)+b*(m-k)+k**2)-b*k*m,3)[1] for a in range(1,k//3) for b in range(a+1,k-a+1>>1)))
A375323_list = list(islice(A375323_gen(),58)) # Chai Wah Wu, Oct 10 2024