|
%I #85 Sep 06 2023 03:07:44
%S 1,2,3,4,-5,6,7,8,9,-10,11,12,-13,14,-15,16,-17,18,19,-20,21,22,23,24,
%T 25,-26,27,28,-29,-30,31,32,33,-34,-35,36,-37,38,-39,-40,-41,42,43,44,
%U -45,46,47,48,49,50,-51,-52,-53,54,-55,56,57,-58,59,-60,-61,62,63,64,65
%N a(n) = (-1)^A083025(n)*n.
%C 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.
%C 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
%C Completely multiplicative because A209661 is. - _Andrew Howroyd_, Aug 04 2018
%D Leonhard Euler, Introductio in analysin infinitorum, 1748.
%H Ray Chandler, <a href="/A209662/b209662.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = n*A209661(n).
%e For n = 10 we have that the 10th row of triangle A207338 is [2, -5] therefore a(10) = 2*(-5) = -10.
%t 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 *)
%o (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
%Y Row products of triangle A207338. Numerators are in A000012. Absolute values give A000027.
%Y Cf. A000796, A002144, A002145, A002808, A083025, A207338, A209661, A209921, A209922.
%K sign,frac,easy,mult
%O 1,2
%A _Omar E. Pol_, Mar 15 2012
%E Formula in sequence name from _M. F. Hasler_, Apr 16 2012
%E a(34) corrected by _Ray Chandler_, Mar 19 2016
|
