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

1,1

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

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Possible denominators are A061742.

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

Sequence in context: A322487 A300735 A196457 * A317348 A144416 A362846

Adjacent sequences: A136367 A136368 A136369 * A136371 A136372 A136373

Keyword

nonn,frac

Author

Alexander Adamchuk, Dec 27 2007

Extensions

Definition corrected by Alexander Adamchuk, Sep 15 2010

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

Status

approved