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For n = 7 we can see below some views of two associated polycubes called "prism of partitions" and "tower of divisors". Both objects contains the same number of cubes (that property is also valid for n >= 1).
Front view of the Partitions Position Lateral view of the
prism of partitions. of 7. of the 1's. of the tower of divisors.
Top view of the
of the tower of divisors.
Figure 1 is a two-dimensional diagram of the partitions of 7. The area of the diagram is A066186(7) = 105. Note that the diagram can be interpreted also as the front view of a right prism whose volumen is 1*7*A000041(7) = 1*7*15 = 105, equaling the volume of the tower of divisors that appears in the figures 4 and 5.
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Accumulate @ Table[4 * PartitionsP[k-1] + 2 * DivisorSigma[1, k + 1], {k, 0, 1, 50}] (* Amiram Eldar, Jul 14 2021 *)
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Accumulate @ Table[4 * PartitionsP[k] + 2 * DivisorSigma[1, k + 1], {k, 0, 50}] (* Amiram Eldar, Jul 14 2021 *)
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