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A239738 Normal distributions from the primes. The contracted sequence of integers generated from the frequencies of the summation of elements of the subsets of the Cartesian product of the natural numbers of ascending prime cardinality. that is, given a number of sets of the natural numbers of ascending modulo P(n+1), the probabilities of generating a given number from the selection of one element from each set form the given sequence. 0
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, 17044, 19005, 20832, 22463, 23842, 24921, 25662, 26039 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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 u-th 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)

LINKS

Table of n, a(n) for n=1..70.

CROSSREFS

Sequence in context: A167595 A179382 A161169 * A058202 A257982 A275705

Adjacent sequences:  A239735 A239736 A239737 * A239739 A239740 A239741

KEYWORD

nonn,tabf,uned,obsc

AUTHOR

Stuart Cooper, Mar 26 2014

EXTENSIONS

Used the normal distribution of the sequence to remove repeated elements greater than their mean.

STATUS

approved

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Last modified August 18 18:19 EDT 2018. Contains 313834 sequences. (Running on oeis4.)