A209662 a(n) = (-1)^A083025(n)*n.

1, 2, 3, 4, -5, 6, 7, 8, 9, -10, 11, 12, -13, 14, -15, 16, -17, 18, 19, -20, 21, 22, 23, 24, 25, -26, 27, 28, -29, -30, 31, 32, 33, -34, -35, 36, -37, 38, -39, -40, -41, 42, 43, 44, -45, 46, 47, 48, 49, 50, -51, -52, -53, 54, -55, 56, 57, -58, 59, -60, -61, 62, 63, 64, 65

Offset

1,2

Comments

Also denominators of an infinite series which is equal to pi, if the numerators are 1, 1, 1,..., 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 arises from an infinite series due to Leonhard Euler which is given by: 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... For another version see A209661.

a(n) = -n if n has an odd number of prime factors of the form 4k+1 (counted with multiplicity), else a(n) = n. - M. F. Hasler, Apr 15 2012

Completely multiplicative because A209661 is. - Andrew Howroyd, Aug 04 2018

References

Leonhard Euler, Introductio in analysin infinitorum, 1748.

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Formula

a(n) = n*A209661(n).

Example

For n = 10 we have that the 10th row of triangle A207338 is [2, -5] therefore a(10) = 2*(-5) = -10.

Mathematica

f[p_, e_] := If[Mod[p, 4] == 1, (-1)^e, 1]; a[n_] := n * Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 06 2023 *)

Prog

(PARI) a(n)={my(f=factor(n)); 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. Numerators are in A000012. Absolute values give A000027.

Cf. A000796, A002144, A002145, A002808, A083025, A207338, A209661, A209921, A209922.

Sequence in context: A001489 A038608 A105811 * A272813 A258070 A258071

Adjacent sequences: A209659 A209660 A209661 * A209663 A209664 A209665

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