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A090942
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n-th arithmetic mean = prime(n).
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3
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2, 4, 9, 13, 27, 23, 41, 33, 55, 83, 51, 103, 89, 69, 103, 143, 155, 95, 175, 147, 113, 205, 171, 227, 289, 201, 155, 215, 165, 229, 547, 255, 329, 205, 489, 221, 373, 385, 319, 407, 419, 263, 611, 279, 373, 289, 763, 787, 419, 327, 433, 545, 345, 781, 581, 593
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OFFSET
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1,1
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COMMENTS
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In other words, this is the unique sequence such that for all n >= 1, (1/n) * Sum_{k=1..n} a(k) = prime(n). - Antti Karttunen, Apr 30 2015
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LINKS
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FORMULA
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a(n) = n*prime(n) - (n-1)*prime(n-1).
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EXAMPLE
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Illustration of initial terms:
Consider a diagram in the first quadrant of the square grid in which the length of the n-th horizontal line segment is equal to the n-th prime, as shown below:
. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
. |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 51|
. |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 83| |
. |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 55| | |
. |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 33| | | |
. |_ _ _ _ _ _ _ _ _ _ _ _ _ 41| | | | |
. |_ _ _ _ _ _ _ _ _ _ _ 23| | | | | |
. |_ _ _ _ _ _ _ 27| | | | | | |
. |_ _ _ _ _ 13| | | | | | | |
. |_ _ _ 9| | | | | | | | |
. |_ _ 4| | | | | | | | | |
. |_2_|_|_ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _ _ _ _ _|_ _|
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a(n) is also the area (or the number of cells) in the n-th region of the diagram. For example: a(4) = 7 + 6 = 13.
(End)
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MATHEMATICA
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Table[If[n==1, 2, nPrime@n -(n-1)Prime[n-1]], {n, 60}] (* Michael De Vlieger, Apr 20 2015 *)
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PROG
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(PARI) vector(60, n, if(n==1, 2, n*prime(n) -(n-1)*prime(n-1))) \\ G. C. Greubel, Feb 04 2019
(Magma) [n le 1 select 2 else n*NthPrime(n) - (n-1)*NthPrime(n-1): n in [1..60]]; // G. C. Greubel, Feb 04 2019
(Sage) [2] + [n*nth_prime(n) - (n-1)*nth_prime(n-1) for n in (2..60)] # G. C. Greubel, Feb 04 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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