A136370 Numerator of 1 - Sum_{k=1..n} (-1)^(k+1)/prime(k)^2.

3, 31, 739, 37111, 4446331, 756766039, 217803584371, 78887714418031, 41637516941042299, 35066922176061410359, 33657455280704707522099, 46117280789485930425170431, 77468081652660425646977758411, 143331051198625503752852285686039

Offset: 1

Author

Alexander Adamchuk, Dec 27 2007

Keywords

• nonn
• frac

Comments

It seems that the denominator of 1 - Sum_{k=1..n} (-1)^(k+1)/prime(k)^2 is A061742(n), which is the square of the product of the first n primes, but this is not immediately obvious. - Petros Hadjicostas, May 14 2020

Examples

			The first few fractions are 3/4, 31/36, 739/900, 37111/44100, 4446331/5336100, 756766039/901800900, ... = A136370/A061742. - _Petros Hadjicostas_, May 14 2020
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..195

Crossrefs

Possible denominators are A061742.

Cf. A024530, A136365, A136366, A136367, A136368, A136369, A136371.

Programs

Mathematica

Table[Numerator[1 - Sum[(-1)^(k+1)/Prime[k]^2, {k, 1, n}]], {n, 1, 20}]

PARI

a(n) = numerator(1 - sum(k=1, n, (-1)^(k+1)/prime(k)^2)); \\ Michel Marcus, May 14 2020

Python

from sympy import prime
from fractions import Fraction
from itertools import accumulate, count, islice
def A136370gen(): yield from map(lambda x: (1-x).numerator, accumulate(Fraction((-1)**(k+1), prime(k)**2) for k in count(1)))
print(list(islice(A136370gen(), 14))) # Michael S. Branicky, Jun 26 2022

Formula

A136370/A061742 tends to 1 - A242301 = 0.83718375333639858423166... - Vaclav Kotesovec, May 14 2020

Extensions

Definition corrected by Alexander Adamchuk, Sep 15 2010

a(14) and beyond from Michael S. Branicky, Jun 26 2022