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A090942 n-th arithmetic mean = prime(n).

%I A090942 #36 Sep 08 2022 08:45:12
%S A090942 2,4,9,13,27,23,41,33,55,83,51,103,89,69,103,143,155,95,175,147,113,
%T A090942 205,171,227,289,201,155,215,165,229,547,255,329,205,489,221,373,385,
%U A090942 319,407,419,263,611,279,373,289,763,787,419,327,433,545,345,781,581,593
%N A090942 n-th arithmetic mean = prime(n).
%C A090942 Partial sums give A033286. - _Omar E. Pol_, Apr 20 2015
%C A090942 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
%H A090942 G. C. Greubel, <a href="/A090942/b090942.txt">Table of n, a(n) for n = 1..10000</a>
%F A090942 a(n) = n*prime(n) - (n-1)*prime(n-1).
%F A090942 a(n) = A033287(n-2), n>1. - _R. J. Mathar_, Sep 08 2008
%F A090942 a(n) = A000040(n) + A141042(n-1), n >=2. - _Omar E. Pol_, Apr 20 2015
%e A090942 From _Omar E. Pol_, Apr 20 2015: (Start)
%e A090942 Illustration of initial terms:
%e A090942 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:
%e A090942 .   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
%e A090942 .  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  51|
%e A090942 .  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _          83|   |
%e A090942 .  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _      55|           |   |
%e A090942 .  |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _  33|       |           |   |
%e A090942 .  |_ _ _ _ _ _ _ _ _ _ _ _ _      41|   |       |           |   |
%e A090942 .  |_ _ _ _ _ _ _ _ _ _ _  23|       |   |       |           |   |
%e A090942 .  |_ _ _ _ _ _ _      27|   |       |   |       |           |   |
%e A090942 .  |_ _ _ _ _  13|       |   |       |   |       |           |   |
%e A090942 .  |_ _ _   9|   |       |   |       |   |       |           |   |
%e A090942 .  |_ _ 4|   |   |       |   |       |   |       |           |   |
%e A090942 .  |_2_|_|_ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _ _ _ _ _|_ _|
%e A090942 .
%e A090942 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.
%e A090942 (End)
%t A090942 Table[If[n==1, 2, nPrime@n -(n-1)Prime[n-1]], {n, 60}] (* _Michael De Vlieger_, Apr 20 2015 *)
%o A090942 (PARI) vector(60, n, if(n==1,2, n*prime(n) -(n-1)*prime(n-1))) \\ _G. C. Greubel_, Feb 04 2019
%o A090942 (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
%o A090942 (Sage) [2] + [n*nth_prime(n) - (n-1)*nth_prime(n-1) for n in (2..60)] # _G. C. Greubel_, Feb 04 2019
%Y A090942 Cf. A000040, A007504, A033286, A033287, A141042, A152535.
%K A090942 nonn
%O A090942 1,1
%A A090942 _Amarnath Murthy_, Dec 29 2003
%E A090942 Corrected and extended by _Ray Chandler_, Dec 31 2003