Which integers can be expressed as $$a^3+b^3+c^3-3abc$$? $$a$$, $$b$$, $$c\in\Bbb Z$$.

$(a\pm1)^3+a^3+a^3-3(a\pm1)a^2=3a\pm1$

$(a-1)^3+a^3+(a+1)^3-3a(a+1)(a-1)=9a$

$2(a^3+b^3+c^3-3abc)=3(a+b+c)(a^2+b^2+c^2)-(a+b+c)^3$

If $$3\mid(a^3+b^3+c^3-3abc)$$, then $$3\mid(a+b+c)^3$$, $$3\mid(a+b+c)$$. so $$9\mid(a^3+b^3+c^3-3abc)$$.

All $$n$$ such that $$3\nmid n$$ or $$9\mid n$$.

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