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A136370
Numerator of 1 - Sum_{k=1..n} (-1)^(k+1)/prime(k)^2.
8
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.
Sequence in context: A322487 A300735 A196457 * A373873 A317348 A144416
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