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

1,1

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.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Junior Balkan Team Selection Tests - Romania 2023, Problems

Example

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.

Maple

g:= proc(q) local R, F, ix, iy, x, y, z, r;
        R:= {}:
        F:= convert(numtheory:-divisors(q), list);
        for ix from 2 to nops(F) do
         x:= F[ix];
         for iy from 1 to ix-1 do
           y:= F[iy];
           z:= q/(x*y);
           if z::integer and nops({x, y, z}) = 3 then
             r:= (x+y+z)/2;
             if r::integer and x + y - z > 0 and x - y + z > 0 and y - x + z > 0 then R:= R union {r} fi
           fi;
        od od;
        R
end proc:
select(`<=`, `union`(seq(g(x^3), x=1..200)), 200); # Robert Israel, Sep 15 2025

Prog

(Magma) [n:n in [1..200]|exists(u){<a, b, n-a-b>: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

Crossrefs

Sequence in context: A287058 A061016 A038366 * A210539 A173231 A172055

Adjacent sequences: A375320 A375321 A375322 * A375324 A375325 A375326

Keyword

nonn

Author

Marius A. Burtea, Sep 15 2024

Status

approved