A301275 Numerator of variance of first n primes.

0, 1, 7, 59, 64, 581, 649, 2287, 1001, 2443, 5669, 17915, 6665, 36637, 3529, 22413, 22813, 13065, 75865, 191819, 58778, 289013, 7627, 141973, 5213, 628001, 370333, 96211, 249436, 381167, 672727, 1565639, 453767, 691587, 1194917, 301867, 770294

Offset: 1

Author

N. J. A. Sloane, Mar 18 2018

Keywords

• nonn
• frac

Comments

Variance here is the sample variance unbiased estimator. - Chai Wah Wu, Mar 22 2018

Examples

			The variances are 0, 1/2, 7/3, 59/12, 64/5, 581/30, 649/21, 2287/56, 1001/18, 2443/30, 5669/55, 17915/132, 6665/39, 36637/182, 3529/15, 22413/80, 22813/68, 13065/34, 75865/171, 191819/380, 58778/105, 289013/462, 7627/11, 141973/184, 5213/6, 628001/650, ...
		

Links

• Chai Wah Wu, 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.

Crossrefs

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

Programs

Maple

v := n -> 1/(n-1) * add((ithprime(i)  add(ithprime(j),j=1..n)/n)^2, i=1..n );
v1:= [0, seq(v(n),n=2..70)];

Mathematica

a[n_] := If[n == 1, 0, Variance[Prime[Range[n]]] // Numerator];
a /@ Range[100] (* Jean-François Alcover, Oct 27 2019 *)

Python

from fractions import Fraction
from sympy import prime
mu, variance = Fraction(prime(1)), Fraction(0)
A301275_list = [variance.numerator]
for i in range(2,10001):
    datapoint = prime(i)
    newmu = mu+(datapoint-mu)/i
    variance = (variance*(i-2) + (datapoint-mu)*(datapoint-newmu))/(i-1)
    mu = newmu
    A301275_list.append(variance.numerator) # Chai Wah Wu, Mar 22 2018