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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.
0
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
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.
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
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Sep 15 2024
STATUS
approved