A069733 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.
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, 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, 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
Offset: 1
Links
- Antti Karttunen, Table of n, a(n) for n = 1..10000
- Valery A. Liskovets and Alexander Mednykh, Number of non-orientable coverings of the Klein bottle, 2002.
Programs
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Mathematica
Table[Count[Divisors[n],?(Mod[#,4]!=0&)],{n,110}] (* _Harvey P. Dale, Jan 10 2016 *) 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 *)
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PARI
a(n)=if(n<1,0,sumdiv(n,d,sign(d%4)))
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PARI
a(n) = my(v = valuation(n, 2)); if(v > 1, n>>=(v-1)); numdiv(n) \\ David A. Corneth, Aug 28 2023
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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
Formula
Multiplicative with a(2^e)=2 and a(p^e)=e+1 for e>0 and an odd prime p.
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).
G.f.: Sum_{m>0} x^m*(1+x^m+x^(2*m))/(1-x^(4*m)). - Vladeta Jovovic, Oct 21 2002
From Amiram Eldar, Dec 05 2022: (Start)
Dirichlet g.f.: zeta(s)^2*(1 - 1/4^s).
Sum_{k=1..n} a(k) ~ (3 * n * log(n) + (6*gamma + 2*log(2) - 3)*n)/4, where gamma is Euler's constant (A001620). [Corrected by Andrey Zabolotskiy, Apr 20 2025] (End)
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