This site is supported by donations to The OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

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