OFFSET
1
COMMENTS
Also numerators of an infinite series which is equal to pi, if the denominators are the natural numbers A000027, for example: pi = 1/1 + 1/2 + 1/3 + 1/4 - 1/5 + 1/6 + 1/7 + 1/8 + 1/9 - 1/10 + 1/11 + 1/12 - 1/13 + 1/14 ... = 3.14159263... This remarkable result is due to Leonhard Euler. For another version see A209662.
REFERENCES
Leonhard Euler, Introductio in analysin infinitorum, 1748.
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A209662(n)/n.
Completely multiplicative with a(p) = -1 for p mod 4 = 1, a(p) = 1 otherwise. - Andrew Howroyd, Aug 04 2018
EXAMPLE
For n = 10 we have that the 10th row of triangle A207338 is [2, -5] therefore a(10) = 2*(-5)/10 = -1.
MATHEMATICA
f[p_, e_] := If[Mod[p, 4] == 1, (-1)^e, 1]; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 06 2023 *)
PROG
(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); if(p%4==1, -1, 1)^e)} \\ Andrew Howroyd, Aug 04 2018
CROSSREFS
KEYWORD
sign,frac,easy,mult
AUTHOR
Omar E. Pol, Mar 15 2012
EXTENSIONS
Formula in sequence name from M. F. Hasler, Apr 16 2012
a(34) corrected by Ray Chandler, Mar 19 2016
STATUS
approved