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A183063 Number of even divisors of n. 41

%I

%S 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,

%T 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,

%U 0,4,0,9,0,2,0,4,0,4,0,8,0,2,0,8,0,2

%N Number of even divisors of n.

%C 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 k-1 zeros followed by the members of A000005, with k-1 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

%C Number of zeros in n-th row of triangle A247795. - _Reinhard Zumkeller_, Sep 28 2014

%C 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

%C a(n) is also the number of partitions of n into an even number of equal parts. - _Omar E. Pol_, May 14 2017

%H Charles R Greathouse IV, <a href="/A183063/b183063.txt">Table of n, a(n) for n = 1..10000</a>

%H Mircea Merca, <a href="http://dx.doi.org/10.1016/j.jnt.2015.08.014">Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer</a>, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_e(n).

%F a(n) = A000005(n) - A001227(n).

%F a(2n-1) = 0; a(2n) = A000005(n).

%F 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

%F 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

%F From _Ridouane Oudra_, Sep 02 2019: (Start)

%F a(n) = Sum_{i=1..n} (floor(n/(2*i)) - floor((n-1)/(2*i))).

%F a(n) = 2*A000005(n) - A000005(2n). (End)

%F 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

%e For n = 12, set of even divisors is {2, 4, 6, 12}, so a(12) = 4.

%e 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

%p A183063 := proc(n)

%p if type(n,'even') then

%p numtheory[tau](n/2) ;

%p else

%p 0;

%p end if;

%p end proc: # _R. J. Mathar_, Jun 18 2015

%t Table[Length[Select[Divisors[n], EvenQ]], {n, 90}] (* _Alonso del Arte_, Jan 10 2012 *)

%o (PARI) a(n)=if(n%2,0,numdiv(n/2)) \\ _Charles R Greathouse IV_, Jul 29 2011

%o (Haskell)

%o a183063 = sum . map (1 -) . a247795_row

%o -- _Reinhard Zumkeller_, Sep 28 2014, Jan 15 2013, Jan 10 2012

%o (Sage)

%o def A183063(n): return len([1 for d in divisors(n) if is_even(d)])

%o [A183063(n) for n in (1..80)] # _Peter Luschny_, Feb 01 2012

%o (MAGMA) [IsOdd(n) select 0 else #[d:d in Divisors(n)|IsEven(d)]:n in [1..100]]; // _Marius A. Burtea_, Dec 16 2019

%Y Cf. A001227, A000593, A183064, A136655, A125911, A320111.

%Y Column 2 of A195050. - _Omar E. Pol_, Oct 19 2011

%Y Cf. A027750, A083910.

%Y Cf. A247795.

%K nonn,easy

%O 1,4

%A _Jaroslav Krizek_, Dec 22 2010

%E Formula corrected by _Charles R Greathouse IV_, Jul 29 2011

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Last modified March 30 18:21 EDT 2020. Contains 333127 sequences. (Running on oeis4.)