Erdős Problem #1107
Let r ≥ 2. Is every large integer the sum of at most r + 1 many r-powerful numbers?
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Open problems about powerful numbers
A number is powerful if every prime dividing it divides it at least twice. Questions ask about consecutive powerful numbers, sums of powerful numbers and their gaps.
Erdős Problem #137
We say that N is powerful if whenever p| N we also have p^2| N. Let k≥ 3. Can the product of any k consecutive positive integers ever be powerful?
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Erdős Problem #364
There is no consecutive triple of powerful numbers.
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Erdős Problem #366
Are there any 2-full n such that n+1 is 3-full?
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Erdős Problem #367
Let B_2(n) be the 2-full part of n (that is, B_2(n)=n/n' where n' is the product of all primes that divide n exactly once). Is it true that, for every fixed k ≥ 1, Π_n ≤ m < n+k B_2(m) ≪ n^2+o(1)?
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Erdős Problem #938
Let A=n_1 < n_2 < ⋯ be the sequence of powerful numbers (if p| n then p^2| n). Are there only finitely many three-term progressions of consecutive terms n_k,n_k+1,n_k+2?
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Erdős Problem #939
If r≥4 then can the sum of r-2 coprime r-powerful numbers ever be itself r-powerful?
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Erdős Problem #940
Let r ≥ 3. Is it true that the set of integers which are the sum of at most r r-powerful numbers has density 0?
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Erdős Problem #942
Is there some constant c > 0 such that h(n) < (log n)^c + o(1) and, for infinitely many n, h(n) > (log n)^c - o(1).
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Erdős Problem #943
Let A be the set of powerful numbers. Is is true that 1_Aast 1_A(n)=n^o(1) for every n?
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