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

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

Author

Amarnath Murthy, Dec 29 2003

Keywords

• nonn

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

Examples

			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)
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A007504, A033286, A033287, A141042, A152535.

Programs

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

Mathematica

Table[If[n==1, 2, nPrime@n -(n-1)Prime[n-1]], {n, 60}] (* Michael De Vlieger, Apr 20 2015 *)

PARI

vector(60, n, if(n==1,2, n*prime(n) -(n-1)*prime(n-1))) \\ 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

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

Extensions

Corrected and extended by Ray Chandler, Dec 31 2003