%I #36 Sep 08 2022 08:45:12

%S 2,4,9,13,27,23,41,33,55,83,51,103,89,69,103,143,155,95,175,147,113,

%T 205,171,227,289,201,155,215,165,229,547,255,329,205,489,221,373,385,

%U 319,407,419,263,611,279,373,289,763,787,419,327,433,545,345,781,581,593

%N n-th arithmetic mean = prime(n).

%C Partial sums give A033286. - _Omar E. Pol_, Apr 20 2015

%C 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 G. C. Greubel, <a href="/A090942/b090942.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n*prime(n) - (n-1)*prime(n-1).

%F a(n) = A033287(n-2), n>1. - _R. J. Mathar_, Sep 08 2008

%F a(n) = A000040(n) + A141042(n-1), n >=2. - _Omar E. Pol_, Apr 20 2015

%e From _Omar E. Pol_, Apr 20 2015: (Start)

%e Illustration of initial terms:

%e 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 . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 51|

%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 83| |

%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 55| | |

%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 33| | | |

%e . |_ _ _ _ _ _ _ _ _ _ _ _ _ 41| | | | |

%e . |_ _ _ _ _ _ _ _ _ _ _ 23| | | | | |

%e . |_ _ _ _ _ _ _ 27| | | | | | |

%e . |_ _ _ _ _ 13| | | | | | | |

%e . |_ _ _ 9| | | | | | | | |

%e . |_ _ 4| | | | | | | | | |

%e . |_2_|_|_ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _|_ _ _ _|_ _ _ _ _ _|_ _|

%e .

%e 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 (End)

%t Table[If[n==1, 2, nPrime@n -(n-1)Prime[n-1]], {n, 60}] (* _Michael De Vlieger_, Apr 20 2015 *)

%o (PARI) vector(60, n, if(n==1,2, n*prime(n) -(n-1)*prime(n-1))) \\ _G. C. Greubel_, Feb 04 2019

%o (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 (Sage) [2] + [n*nth_prime(n) - (n-1)*nth_prime(n-1) for n in (2..60)] # _G. C. Greubel_, Feb 04 2019

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

%K nonn

%O 1,1

%A _Amarnath Murthy_, Dec 29 2003

%E Corrected and extended by _Ray Chandler_, Dec 31 2003