login
A090942
n-th arithmetic mean = prime(n).
3
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
OFFSET
1,1
COMMENTS
Partial sums give A033286. - Omar E. Pol, Apr 20 2015
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
LINKS
FORMULA
a(n) = n*prime(n) - (n-1)*prime(n-1).
a(n) = A033287(n-2), n>1. - R. J. Mathar, Sep 08 2008
a(n) = A000040(n) + A141042(n-1), n >=2. - Omar E. Pol, Apr 20 2015
EXAMPLE
From Omar E. Pol, Apr 20 2015: (Start)
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_|_|_ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _ _ _ _ _|_ _|
.
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)
MATHEMATICA
Table[If[n==1, 2, nPrime@n -(n-1)Prime[n-1]], {n, 60}] (* Michael De Vlieger, Apr 20 2015 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Amarnath Murthy, Dec 29 2003
EXTENSIONS
Corrected and extended by Ray Chandler, Dec 31 2003
STATUS
approved