2014-11-08-082031.sagews

@interact
def f(n=4):
    a = (1+1/n)^n
    print N(a,1000)
    print N(e,1000)

Interact: please open in CoCalc

for X in [500000,1000000,..,3000000]:
    print X, prime_pi(X)

500000 41538 1000000 78498 1500000 114155 2000000 148933 2500000 183072 3000000 216816

prime_range(10010)
︠81ef287d-617b-431f-954b-afa2e8ab3660︠
show(sum(point3d([p/log(p),log(p),p], size=5) for p in primes(100)), frame=False)
︠3bba7d3c-6107-4a4a-9602-e8c815eb89d4︠
%python
def err(X):
    n = 0
    for p in prime_range(X):
        n += 1
        if random() <= 0.01:
            if random() <=0.5:
                n -= 1
            else:
                n += 1

    return n - prime_pi(X)

%time err(X)

-61 CPU time: 0.18 s, Wall time: 0.18 s

X = 3000000
%time v = stats.TimeSeries([abs(err(X)) for i in range(100)])

CPU time: 18.68 s, Wall time: 18.68 s

︠f17e69d0-5dd5-4402-b57e-629f60f14d1a︠
v.plot_histogram()

v.mean(), N(sqrt(X))

(35.61, 1732.05080756888)