$\sum_{k=1}^n a_k^3 = \left(\sum_{k=1}^n a_k \right)^2$

We know that $\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$. Interestingly, $1^3+2^3+2^3+4^3=(1+2+2+4)^2$. Are there other non-consecutive numbers $a_1, a_2, \ldots, a_k$ such that $$\sum_{k=1}^n a_k^3 = \left(\sum_{k=1}^n a_k \right)^2?$$

Solution 1:

Here is a surprising result: $$ \sum_{d\mid n} \tau(d)^3 = \big(\sum_{d\mid n} \tau(d)\big)^2 $$ where $\tau$ counts the number of divisors of an integer.

It's exercise 12 in chapter 2 of Apostol's Introduction to Analytic Number Theory.

If you take $n=2^N$, you get the classic result for consecutive numbers.

If you take $n=pq$, a product of two primes, you get your example.

I don't think all examples come from the result above, though.

See John Mason, Generalising 'Sums of Cubes Equal to Squares of Sums', The Mathematical Gazette, Vol. 85, No. 502 (Mar., 2001), pp. 50-58.

Solution 2:

Lots. Here's some examples:

[ 2, 2 ]
[ 3, 3, 3 ]
[ 1, 2, 2, 4 ]
[ 2, 2, 4, 4 ]
[ 4, 4, 4, 4 ]
[ 1, 2, 2, 3, 5 ]
[ 3, 3, 3, 3, 6 ]
[ 3, 3, 3, 4, 6 ]
[ 5, 5, 5, 5, 5 ]
[ 1, 1, 1, 2, 2, 5 ]
[ 1, 1, 1, 4, 4, 5 ]
[ 1, 1, 2, 4, 5, 5 ]
[ 1, 1, 4, 5, 5, 5 ]
[ 1, 2, 2, 3, 4, 6 ]
[ 1, 2, 2, 4, 4, 6 ]
[ 1, 4, 4, 4, 6, 6 ]
[ 2, 2, 2, 2, 2, 6 ]
[ 2, 2, 4, 4, 6, 6 ]
[ 2, 4, 4, 5, 5, 7 ]
[ 2, 4, 4, 6, 6, 6 ]
[ 3, 3, 3, 3, 5, 7 ]
[ 3, 3, 3, 6, 6, 6 ]
[ 3, 4, 5, 5, 6, 7 ]
[ 3, 5, 5, 5, 6, 7 ]
[ 4, 5, 5, 6, 6, 7 ]
[ 6, 6, 6, 6, 6, 6 ]
[ 1, 1, 2, 2, 5, 5, 6 ]
[ 1, 1, 3, 4, 4, 5, 7 ]
[ 1, 1, 4, 5, 5, 5, 7 ]
[ 1, 1, 5, 5, 6, 6, 6 ]
[ 1, 2, 2, 2, 3, 6, 6 ]
[ 1, 2, 2, 3, 3, 3, 7 ]
[ 1, 2, 2, 3, 4, 5, 7 ]
[ 1, 2, 2, 4, 6, 6, 6 ]
[ 1, 2, 3, 6, 6, 6, 6 ]
[ 1, 2, 6, 6, 6, 6, 6 ]
[ 1, 3, 5, 5, 5, 7, 7 ]
[ 1, 4, 4, 5, 6, 7, 7 ]
[ 1, 5, 5, 6, 6, 7, 7 ]
[ 2, 2, 2, 2, 5, 5, 7 ]
[ 2, 2, 4, 4, 4, 4, 8 ]
[ 2, 3, 3, 3, 4, 4, 8 ]
[ 2, 3, 3, 3, 5, 7, 7 ]
[ 2, 3, 3, 5, 6, 7, 7 ]
[ 2, 3, 6, 6, 6, 7, 7 ]
[ 2, 4, 4, 6, 6, 6, 8 ]
[ 2, 6, 6, 6, 6, 6, 8 ]
[ 3, 3, 3, 3, 4, 6, 8 ]
[ 3, 3, 3, 3, 5, 6, 8 ]
[ 3, 3, 3, 4, 6, 6, 8 ]
[ 3, 5, 5, 5, 7, 7, 8 ]
[ 4, 4, 4, 5, 5, 5, 9 ]
[ 4, 5, 5, 5, 6, 6, 9 ]
[ 6, 6, 6, 6, 6, 6, 9 ]
[ 7, 7, 7, 7, 7, 7, 7 ]
[ 1, 1, 1, 1, 1, 1, 5, 5 ]
[ 1, 1, 1, 1, 1, 2, 3, 6 ]
[ 1, 1, 1, 1, 2, 2, 5, 6 ]
[ 1, 1, 1, 1, 4, 5, 6, 6 ]
[ 1, 1, 1, 2, 2, 5, 6, 6 ]
[ 1, 1, 1, 2, 5, 5, 5, 7 ]
[ 1, 1, 1, 2, 5, 6, 6, 6 ]
[ 1, 1, 2, 2, 2, 2, 4, 7 ]
[ 1, 1, 2, 2, 3, 5, 6, 7 ]
[ 1, 1, 2, 4, 4, 5, 5, 8 ]
[ 1, 1, 2, 4, 5, 5, 5, 8 ]
[ 1, 1, 3, 3, 3, 4, 5, 8 ]
[ 1, 1, 4, 5, 5, 5, 7, 8 ]
[ 1, 2, 2, 2, 4, 4, 4, 8 ]
[ 1, 2, 2, 2, 4, 5, 7, 7 ]
[ 1, 2, 2, 2, 5, 5, 7, 7 ]
[ 1, 2, 2, 3, 3, 3, 7, 7 ]
[ 1, 2, 2, 3, 4, 4, 6, 8 ]
[ 1, 2, 2, 3, 4, 5, 6, 8 ]
[ 1, 2, 2, 4, 4, 6, 6, 8 ]
[ 1, 2, 2, 5, 5, 7, 7, 7 ]
[ 1, 2, 3, 3, 4, 7, 7, 7 ]
[ 1, 2, 4, 5, 5, 7, 7, 8 ]
[ 1, 2, 5, 6, 6, 7, 7, 8 ]
[ 1, 2, 5, 7, 7, 7, 7, 7 ]
[ 1, 3, 3, 3, 3, 5, 7, 8 ]
[ 1, 3, 3, 3, 6, 6, 7, 8 ]
[ 1, 3, 4, 4, 4, 5, 8, 8 ]
[ 1, 3, 4, 4, 5, 6, 8, 8 ]
[ 1, 3, 4, 6, 6, 6, 8, 8 ]
[ 1, 3, 4, 6, 7, 7, 7, 8 ]
[ 1, 3, 5, 5, 5, 5, 7, 9 ]
[ 1, 4, 4, 6, 6, 6, 7, 9 ]
[ 1, 4, 5, 5, 7, 7, 8, 8 ]
[ 1, 4, 5, 6, 7, 7, 8, 8 ]
[ 1, 5, 5, 7, 7, 7, 8, 8 ]
[ 1, 6, 6, 6, 6, 8, 8, 8 ]
[ 2, 2, 2, 2, 3, 3, 3, 8 ]
[ 2, 2, 2, 2, 3, 6, 7, 7 ]
[ 2, 2, 2, 3, 5, 5, 7, 8 ]
[ 2, 2, 2, 3, 6, 7, 7, 7 ]
[ 2, 2, 2, 5, 7, 7, 7, 7 ]
[ 2, 2, 3, 4, 4, 4, 5, 9 ]
[ 2, 2, 3, 4, 4, 7, 7, 8 ]
[ 2, 2, 3, 4, 6, 7, 7, 8 ]
[ 2, 2, 4, 4, 4, 4, 8, 8 ]
[ 2, 2, 4, 4, 4, 6, 6, 9 ]
[ 2, 2, 4, 4, 6, 6, 6, 9 ]
[ 2, 2, 4, 4, 6, 6, 8, 8 ]
[ 2, 2, 5, 5, 6, 7, 8, 8 ]
[ 2, 2, 5, 7, 7, 7, 7, 8 ]
[ 2, 2, 6, 7, 7, 7, 7, 8 ]
[ 2, 3, 3, 5, 5, 6, 7, 9 ]
[ 2, 3, 4, 5, 5, 7, 7, 9 ]
[ 2, 3, 4, 6, 7, 7, 8, 8 ]
[ 2, 3, 6, 7, 7, 7, 8, 8 ]
[ 2, 4, 4, 6, 6, 6, 8, 9 ]
[ 2, 4, 6, 6, 6, 7, 8, 9 ]
[ 2, 5, 5, 5, 5, 5, 6, 10 ]
[ 2, 5, 5, 6, 7, 7, 8, 9 ]
[ 2, 6, 6, 6, 6, 6, 6, 10 ]
[ 2, 6, 6, 6, 8, 8, 8, 8 ]
[ 2, 7, 7, 7, 7, 8, 8, 8 ]
[ 3, 3, 3, 3, 3, 6, 6, 9 ]
[ 3, 3, 3, 3, 4, 5, 7, 9 ]
[ 3, 3, 3, 3, 5, 6, 7, 9 ]
[ 3, 3, 3, 3, 5, 7, 8, 8 ]
[ 3, 3, 4, 4, 5, 6, 8, 9 ]
[ 3, 3, 5, 7, 7, 8, 8, 8 ]
[ 3, 4, 5, 5, 5, 6, 7, 10 ]
[ 3, 4, 5, 5, 6, 6, 7, 10 ]
[ 3, 4, 5, 6, 6, 8, 8, 9 ]
[ 3, 4, 7, 7, 7, 7, 8, 9 ]
[ 3, 5, 5, 5, 6, 7, 7, 10 ]
[ 3, 5, 6, 6, 6, 7, 9, 9 ]
[ 3, 6, 6, 6, 7, 7, 7, 10 ]
[ 3, 6, 7, 7, 7, 8, 8, 9 ]
[ 4, 4, 4, 4, 4, 4, 6, 10 ]
[ 4, 4, 4, 4, 5, 5, 7, 10 ]
[ 4, 4, 4, 4, 8, 8, 8, 8 ]
[ 4, 4, 4, 5, 5, 5, 9, 9 ]
[ 4, 4, 4, 5, 5, 6, 9, 9 ]
[ 4, 4, 5, 5, 5, 7, 9, 9 ]
[ 4, 4, 6, 6, 7, 7, 9, 9 ]
[ 4, 5, 5, 5, 8, 8, 8, 9 ]
[ 4, 5, 5, 6, 6, 7, 8, 10 ]
[ 4, 6, 6, 6, 7, 8, 9, 9 ]
[ 5, 5, 5, 7, 7, 7, 8, 10 ]
[ 5, 5, 7, 7, 7, 8, 9, 9 ]
[ 6, 6, 6, 6, 6, 6, 9, 10 ]
[ 6, 6, 7, 7, 8, 8, 9, 9 ]
[ 6, 6, 8, 8, 8, 8, 8, 9 ]
[ 8, 8, 8, 8, 8, 8, 8, 8 ]

This was generated by the GAP code:

for s in [1..8] do
  A:=UnorderedTuples([1..20],s);;
  for L in A do
    lhs:=Sum(List(L,n->n^3));
    rhs:=Sum(L)^2;
    if(lhs=rhs and L<>[1..s]) then Print(L,"\n"); fi;
  od;
od;