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

Author

Omar E. Pol, Mar 15 2012

Keywords

• sign
• frac
• easy
• mult

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

Examples

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

References

• Leonhard Euler, Introductio in analysin infinitorum, 1748.

Links

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

Crossrefs

Row products of triangle A207338. Numerators are in A000012. Absolute values give A000027.

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

Programs

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 *)

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

Formula

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

Extensions

Formula in sequence name from M. F. Hasler, Apr 16 2012

a(34) corrected by Ray Chandler, Mar 19 2016