A062049 -id:A062049 - cp's OEIS frontend

A233824 A recurrent sequence in Panaitopol's formula for pi(x), where pi(x) is the number of primes <= x.

0, 1, 3, 13, 71, 461, 3447, 29093, 273343, 2829325, 31998903, 392743957, 5201061455, 73943424413, 1123596277863, 18176728317413, 311951144828863, 5661698774848621, 108355864447215063, 2181096921557783605

Offset: 0

Jonathan Sondow, Dec 17 2013

nonn

Sum_{k=0..n} k!*a(n-k) = n*n!.

Panaitopol proved that x/pi(x) = log(x) - 1 - Sum_{k=1..m} a(k)/log(x)^k + O(1/log(x)^{m+1}) for m > 0.

			0!*a(0) = a(0) = 0*0!, so a(0) = 0.
0!*a(1) + 1!*a(0) = a(1) + a(0) = 1*1!, so a(1) = 1.
0!*a(2) + 1!*a(1) + 2!*a(0) = a(2) + a(1) + 2*a(0) = 2*2!, so a(2) = 4 - 1 = 3.

G. C. Greubel, Table of n, a(n) for n = 0..445
Safia Aoudjit and Djamel Berkane, Explicit Estimates Involving the Primorial Integers and Applications, J. Int. Seq., Vol. 24 (2021), Article 21.7.8.
M. Hassani, Some remarks on a determinant related to the prime counting function, The 8th Seminar on Linear Algebra and its Applications, 13-14th May 2015, University of Kurdistan, Iran.
A. Ivić, Review of "A formula for pi(x) applied to a result of Koninck-Ivić" by L. Panaitopol, Zbl 0982.11003.
A. Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht's Conjecture, J. Integer Sequences, 18 (2015), Article 15.11.2. arXiv:1506.03042
A. Kourbatov, On the geometric mean of the first n primes, arXiv:1603.00855 [math.NT], 2016.
L. Panaitopol, A formula for pi(x) applied to a result of Koninck-Ivić, Nieuw Arch. Wiskd., (5) 1, No. 1 (2000), 55-56.

Cf. A000720, A003319, A062049.

a[0] = 0; a[n_] := a[n] = n*n! - Sum[ k! a[n - k], {k, n - 1}]; Table[a@ n, {n, 0, 19}] (* Michael De Vlieger, Mar 26 2016 *)

a(n) = n*n! - Sum_{k=1..n-1} k!*a(n-k).

a(n) = A003319(n+1) if n > 0. (Proof. Set b(n) = A003319(n), so that b(n) = n! - Sum_{k=1..n-1} k!*b(n-k). To get b(n+1) = a(n) for n > 0, induct on n, use (n+1)! = n*n! + n!, and replace k with k+1 in the sum.)

A301277 Nearest integer to mean of first n primes.

Original entry on oeis.org

2, 3, 3, 4, 6, 7, 8, 10, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 45, 47, 49, 51, 53, 55, 58, 60, 63, 65, 67, 70, 72, 75, 77, 80, 82, 85, 87, 90, 92, 94, 97, 100, 102, 105, 107, 110, 113, 115, 118, 121, 123, 126, 128, 131, 133

Offset: 1

N. J. A. Sloane, Mar 18 2018

nonn

Differs from A075465 where ties are involved. - R. J. Mathar, Mar 20 2018

			The means are 2, 5/2, 10/3, 17/4, 28/5, 41/6, 58/7, 77/8, 100/9, 129/10, 160/11, 197/12, 238/13, 281/14, 328/15, 381/16, 440/17, 167/6, 568/19, 639/20, 712/21, 791/22, 38, 321/8, 212/5, ...

Altug Alkan, Table of n, a(n) for n = 1..10000
Joel E. Cohen, Statistics of Primes (and Probably Twin Primes) Satisfy Taylor's Law from Ecology, The American Statistician, 70 (2016), 399-404.

Mean and variance of primes: A301273/A301274, A301275/A301276, A301277, A273462.

Cf. A007504, A060620, A062049, A075465, A225804, A250130/A024451.

m := n -> add(ithprime(j),j=1..n)/n;
m1:=[seq(m(n),n=1..100)];
m2:=map(numer,m1); # A301273
m3:=map(denom,m1); # A301274
m4:=map(round,m1); # A301277

Rest@ FoldList[{Append[First@ #1, #2], If[And[EvenQ@ #1, #2 == 1/2] & @@ {IntegerPart@ #, FractionalPart@ #}, Round@ # + 1, Round@ #] &@ Mean@ First@ #1} &, {{2}, 2}, Prime@ Range[2, 63]][[All, -1]] (* Michael De Vlieger, Apr 05 2018 *)

a(n) = round(sum(i=1, n, prime(i))/n); \\ Altug Alkan, Mar 22 2018

a(n) = round(A007504(n) / n). - David A. Corneth, Mar 22 2018

a(n) ~ prime(n)/2 ~ n*log(n)/2. - Daniel Forgues, Mar 22 2018

cp's OEIS Frontend

A233824 A recurrent sequence in Panaitopol's formula for pi(x), where pi(x) is the number of primes <= x.

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