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A069733
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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.
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9
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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
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OFFSET
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1,2
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COMMENTS
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Also number of divisors of n that are not divisible by 4. - Vladeta Jovovic, Dec 16 2002
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LINKS
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FORMULA
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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
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) - 1)*n))/4, where gamma is Euler's constant (A001620). (End)
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MATHEMATICA
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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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PROG
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(PARI) a(n)=if(n<1, 0, sumdiv(n, d, sign(d%4)))
(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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CROSSREFS
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KEYWORD
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mult,easy,nonn
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AUTHOR
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STATUS
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approved
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