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

1,2

Comments

Also number of divisors of n that are not divisible by 4. - Vladeta Jovovic, Dec 16 2002

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.

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) - 1)*n))/4, where gamma is Euler's constant (A001620). (End)

a(n) = A000005(A259445(n)). - David A. Corneth, Aug 28 2023

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 *)

Prog

(PARI) a(n)=if(n<1, 0, sumdiv(n, d, sign(d%4)))
(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
(PARI) a(n) = my(v = valuation(n, 2)); if(v > 1, n>>=(v-1)); numdiv(n) \\ David A. Corneth, Aug 28 2023

Crossrefs

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

Sequence in context: A023157 A228849 A359541 * A187467 A081755 A366538

Adjacent sequences: A069730 A069731 A069732 * A069734 A069735 A069736

Keyword

mult,easy,nonn

Author

Valery A. Liskovets, Apr 07 2002

Status

approved