Pairs of numbers for which sum of their Ramanujan's partitions yields square ?

[Alexander Povolotsky]

Dear SeqFans,

Are there infinite or finite set of pairs of numbers,
for which the sum of their Ramanujan's partitions yields square number ?
Could the sequence (some what analogous to  A000217 and A135405) be
constructed ?

Similarly, are there infinite or finite set of pairs of numbers,
for which the sum of their Ramanujan's partitions yields cubes ?

Thanks,
Alex
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