

A183063


Number of even divisors of n.


41



0, 1, 0, 2, 0, 2, 0, 3, 0, 2, 0, 4, 0, 2, 0, 4, 0, 3, 0, 4, 0, 2, 0, 6, 0, 2, 0, 4, 0, 4, 0, 5, 0, 2, 0, 6, 0, 2, 0, 6, 0, 4, 0, 4, 0, 2, 0, 8, 0, 3, 0, 4, 0, 4, 0, 6, 0, 2, 0, 8, 0, 2, 0, 6, 0, 4, 0, 4, 0, 4, 0, 9, 0, 2, 0, 4, 0, 4, 0, 8, 0, 2, 0, 8, 0, 2
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OFFSET

1,4


COMMENTS

Number of divisors of n that are divisible by 2. More generally, it appears that the sequence formed by starting with an initial set of k1 zeros followed by the members of A000005, with k1 zeros between every one of them, can be defined as "the number of divisors of n that are divisible by k", (k >= 1). For example if k = 1 we have A000005 by definition, if k = 2 we have this sequence. Note that if k >= 3 the sequences are not included in the OEIS because the usual OEIS policy is not to include sequences like this where alternate terms are zero; this is an exception.  Omar E. Pol, Oct 18 2011
Number of zeros in nth row of triangle A247795.  Reinhard Zumkeller, Sep 28 2014
a(n) is also the number of partitions of n into equal parts, minus the number of partitions of n into consecutive parts.  Omar E. Pol, May 04 2017
a(n) is also the number of partitions of n into an even number of equal parts.  Omar E. Pol, May 14 2017


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 6075, function tau_e(n).


FORMULA

a(n) = A000005(n)  A001227(n).
a(2n1) = 0; a(2n) = A000005(n).
G.f.: Sum_{d>=1} x^(2*d)/(1  x^(2*d)) and generally for the number of divisors that are divisible by k: Sum_{d>=1} x^(k*d)/(1  x^(k*d)).  Geoffrey Critzer, Apr 15 2014
Dirichlet g.f.: zeta(s)^2/2^s and generally for the number of divisors that are divisible by k: zeta(s)^2/k^s.  Geoffrey Critzer, Mar 28 2015
From Ridouane Oudra, Sep 02 2019: (Start)
a(n) = Sum_{i=1..n} (floor(n/(2*i))  floor((n1)/(2*i))).
a(n) = 2*A000005(n)  A000005(2n). (End)
Conjecture: a(n) = lim_{x>n} f(Pi*x), where f(x) = sin(x)*Sum_{k>0} (cot(x/(2*k))/(2*k)  1/x).  Velin Yanev, Dec 16 2019


EXAMPLE

For n = 12, set of even divisors is {2, 4, 6, 12}, so a(12) = 4.
On the other hand, there are six partitions of 12 into equal parts: [12], [6, 6], [4, 4, 4], [3, 3, 3, 3], [2, 2, 2, 2, 2, 2] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. And there are two partitions of 12 into consecutive parts: [12] and [5, 4, 3], so a(12) = 6  2 = 4, equaling the number of even divisors of 12.  Omar E. Pol, May 04 2017


MAPLE

A183063 := proc(n)
if type(n, 'even') then
numtheory[tau](n/2) ;
else
0;
end if;
end proc: # R. J. Mathar, Jun 18 2015


MATHEMATICA

Table[Length[Select[Divisors[n], EvenQ]], {n, 90}] (* Alonso del Arte, Jan 10 2012 *)


PROG

(PARI) a(n)=if(n%2, 0, numdiv(n/2)) \\ Charles R Greathouse IV, Jul 29 2011
(Haskell)
a183063 = sum . map (1 ) . a247795_row
 Reinhard Zumkeller, Sep 28 2014, Jan 15 2013, Jan 10 2012
(Sage)
def A183063(n) : return len(filter(is_even, divisors(n)))
[A183063(n) for n in (1..80)] # Peter Luschny, Feb 01 2012
(MAGMA) [IsOdd(n) select 0 else #[d:d in Divisors(n)IsEven(d)]:n in [1..100]]; // Marius A. Burtea, Dec 16 2019


CROSSREFS

Cf. A001227, A000593, A183064, A136655, A125911, A320111.
Column 2 of A195050.  Omar E. Pol, Oct 19 2011
Cf. A027750, A083910.
Cf. A247795.
Sequence in context: A332447 A132747 A301979 * A318979 A172441 A053399
Adjacent sequences: A183060 A183061 A183062 * A183064 A183065 A183066


KEYWORD

nonn,easy


AUTHOR

Jaroslav Krizek, Dec 22 2010


EXTENSIONS

Formula corrected by Charles R Greathouse IV, Jul 29 2011


STATUS

approved



