%I #36 Aug 28 2023 08:26:24

%S 1,2,2,2,2,4,2,2,3,4,2,4,2,4,4,2,2,6,2,4,4,4,2,4,3,4,4,4,2,8,2,2,4,4,

%T 4,6,2,4,4,4,2,8,2,4,6,4,2,4,3,6,4,4,2,8,4,4,4,4,2,8,2,4,6,2,4,8,2,4,

%U 4,8,2,6,2,4,6,4,4,8,2,4,5,4,2,8,4,4,4,4,2,12,4,4,4,4,4,4,2,6,6,6,2,8,2,4

%N Number of divisors d of n such that d or n/d is odd. Number of non-orientable coverings of the Klein bottle with n lists.

%C Also number of divisors of n that are not divisible by 4. - _Vladeta Jovovic_, Dec 16 2002

%H Antti Karttunen, <a href="/A069733/b069733.txt">Table of n, a(n) for n = 1..10000</a>

%H Valery A. Liskovets and Alexander Mednykh, <a href="http://www.researchgate.net/publication/251203042_The_Number_of_Non-Orientable_Coverings_of_the_Klein_Bottle">Number of non-orientable coverings of the Klein bottle</a>, 2002.

%F Multiplicative with a(2^e)=2 and a(p^e)=e+1 for e>0 and an odd prime p.

%F a(n) = d(n)-d(n/4) for 4|n and =d(n) otherwise where d(n) is the number of divisors of n (A000005).

%F G.f.: Sum_{m>0} x^m*(1+x^m+x^(2*m))/(1-x^(4*m)). - _Vladeta Jovovic_, Oct 21 2002

%F From _Amiram Eldar_, Dec 05 2022: (Start)

%F Dirichlet g.f.: zeta(s)^2*(1 - 1/4^s).

%F Sum_{k=1..n} a(k) ~ (3 * n * log(n) + (6*gamma + 2*log(2) - 1)*n))/4, where gamma is Euler's constant (A001620). (End)

%F a(n) = A000005(A259445(n)). - _David A. Corneth_, Aug 28 2023

%t Table[Count[Divisors[n],_?(Mod[#,4]!=0&)],{n,110}] (* _Harvey P. Dale_, Jan 10 2016 *)

%t f[2, e_] := 2; f[p_, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 15 2020 *)

%o (PARI) a(n)=if(n<1,0,sumdiv(n,d,sign(d%4)))

%o (Scheme, with memoization-macro definec) (definec (A069733 n) (cond ((= 1 n) n) ((even? n) (* 2 (A069733 (A000265 n)))) (else (* (+ 1 (A067029 n)) (A069733 (A028234 n)))))) ;; _Antti Karttunen_, Sep 23 2017

%o (PARI) a(n) = my(v = valuation(n, 2)); if(v > 1, n>>=(v-1)); numdiv(n) \\ _David A. Corneth_, Aug 28 2023

%Y Cf. A000005, A001620, A046897, A069184, A259445.

%K mult,easy,nonn

%O 1,2

%A _Valery A. Liskovets_, Apr 07 2002