
COMMENTS

Although this sequence initially appears similar to A131791, its derivation is entirely different & it deviates quickly.
By sets of natural numbers of ascending prime cardinality, it is meant
{ℕ_1} = {1,2}, {ℕ_2} = {1,2,3}, {ℕ_3} = {1,2,3,4,5}, {ℕ_4} = {1,2,3,4,5,6,7} ,..., {ℕ_w} = [1,2,3,...,p_w] , p ∈ {P}
with Cartesian products
{ℕ_1} X {ℕ_2} = {1,2} X {1,2,3} = {(1,1),(1,2),(1,3),(2,1),(2,2),(2,3))}, etc.
& the sum of the elements of the product's subsets denoted
Sum[{ℕ_1} X {ℕ_2}] = {(2),(3),(4),(3),(4),(5)}
whose elements have the frequencies [1,2,2,1] (with respect to magnitude). It is these frequencies that form the sequence, the symmetry allows for the omission of repeated terms, & hence the following contraction halves the data without loss of information
[1,1] > [1]
[1,2,2,1] > [1,2]
[1,3,5,6,6,5,3,1] > [1,3,5,6]
& so forth.
When arranged by row of the number of sets used, then P(S_{u,r} = u + r  1) = a_{u,r}/Prod p_r
where the a_{u,r} denotes the uth element of row r, P(S = X) the probability that the sum S equals the value X, and Prod p_r is the product
of the first r primes, then the structure &and symmetry become more apparent.
1
1,2
1,3,5,6
1,4,9,15,21,26,29
1,5,14,29,50,76,105,134,160,181,196,204
1,6,20,49,99,175,280,414,574,755,951,1155,1359,1554,1730,1876,1981,2036
1,7,27,76,175,350,630,1044,1618,2373,3324,4479,5838,7392,9122,10998,
12979,15014,1704,19005,20832,22463,23842,24921,25662,26039
where each row contains 𝓁ᵣ = ½(∑pᵣ  r + 1) terms & clearly the sum of each row must equal half the product of the primes used,
∑aᵤ,ᵣ = ½∏pᵣ. [1 ≤ u ≤ 𝓁ᵣ]
& one can see that in general
for all u, r, P(Sᵤ,ᵣ = r) = P(Sᵤ,ᵣ = ∑pᵣ) = 1/∏pᵣ
P(Sᵤ,ᵣ = r + 1) = P(Sᵤ,ᵣ = ∑pᵣ  1) = r/∏pᵣ
P(S = r + i) = P(S = ∑pᵣ  i) = a(u+i,r)/∏pᵣ ,[0 ≤ i ≤ 𝓁ᵣ  1)]
Sᵤ,ᵣ ~ 𝒩(μᵣ , σᵣ²)
