A259445 -id:A259445 - cp's OEIS frontend

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

Valery A. Liskovets, Apr 07 2002

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

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.

Cf. A069734, A000005, A001620, A046897, A069184, A259445.

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 *)

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

a(n) = my(v = valuation(n, 2)); if(v > 1, n>>=(v-1)); numdiv(n) \\ David A. Corneth, Aug 28 2023

;; 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

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

A287957 Table read by antidiagonals: T(n, k) = greatest common recursive divisor of n and k; n > 0 and k > 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 2, 5, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2, 1, 6, 1, 2, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 7, 2, 1

Offset: 1

Rémy Sigrist, Jun 03 2017

We use the definition of recursive divisor given in A282446.
More informally, the prime tower factorization of T(n, k) is the intersection of the prime tower factorizations of n and k (the prime tower factorization of a number is defined in A182318).
This sequence has connections with the classical GCD (A003989).
For any i > 0, j > 0 and k > 0:
- T(i, j) = 1 iff gcd(i, j) = 1,
- A007947(T(i, j)) = A007947(gcd(i, j)),
- T(i, j) >= 1,
- T(i, j) <= min(i, j),
- T(i, j) <= gcd(i, j),
- T(i, 1) = 1,
- T(i, i) = i,
- T(i, j) = T(j, i) (the sequence is commutative),
- T(i, T(j, k)) = T(T(i, j), k) (the sequence is associative),
- T(i, i*j) <= i,
- if gcd(i, j) = 1 then T(i*j, k) = T(i, k) * T(j, k) (the sequence is multiplicative),
- T(i, 2*i) = A259445(i).
See also A287958 for the LCM equivalent.

			Table starts:
n\k|    1   2   3   4   5   6   7   8   9   10
---+-----------------------------------------------
1  |    1   1   1   1   1   1   1   1   1    1  ...
2  |    1   2   1   2   1   2   1   2   1    2  ...
3  |    1   1   3   1   1   3   1   1   3    1  ...
4  |    1   2   1   4   1   2   1   2   1    2  ...
5  |    1   1   1   1   5   1   1   1   1    5  ...
6  |    1   2   3   2   1   6   1   2   3    2  ...
7  |    1   1   1   1   1   1   7   1   1    1  ...
8  |    1   2   1   2   1   2   1   8   1    2  ...
9  |    1   1   3   1   1   3   1   1   9    1  ...
10 |    1   2   1   2   5   2   1   2   1   10  ...
...
T(4, 8) = T(2^2, 2^3) = 2.

Rémy Sigrist, First 100 antidiagonals of array, flattened
Rémy Sigrist, Illustration of the first terms

Cf. A003989, A007947, A182318, A259445, A282446, A287958.

T(n,k) = my (g=factor(gcd(n,k))); return (prod(i=1, #g~, g[i,1]^T(valuation(n, g[i,1]), valuation(k, g[i,1]))))

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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.

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A287957 Table read by antidiagonals: T(n, k) = greatest common recursive divisor of n and k; n > 0 and k > 0.

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