A209661 a(n) = (-1)^A083025(n).

1, 1, 1, 1, -1, 1, 1, 1, 1, -1, 1, 1, -1, 1, -1, 1, -1, 1, 1, -1, 1, 1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, -1, 1, -1, 1, -1, -1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, -1, 1, -1, -1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, 1, -1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, 1

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

Row products of triangle A207338 divided by n. Absolute values give A000012.

Cf. A000027, A000796, A002144, A002145, A083025, A207338, A209662, A209921, A209922.

Sequence in context: A306638 A076479 A155040 * A033999 A000012 A216430

Adjacent sequences: A209658 A209659 A209660 * A209662 A209663 A209664

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