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%I #107 Dec 16 2023 15:00:07
%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
%F a(n) = A000005(A000265(n))*A007814(n) - _Chai Wah Wu_, Jul 16 2022
%F Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma - 1/2 - log(2)/2)*n, where gamma is Euler's constant (A001620). - _Amiram Eldar_, Nov 27 2022
%F Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = 1-log(2) (A244009). - _Amiram Eldar_, Mar 01 2023
%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 *)
%t a[n_] := (e = IntegerExponent[n, 2]) * DivisorSigma[0, n / 2^e]; Array[a, 100] (* _Amiram Eldar_, Jul 06 2022 *)
%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
%o (Python)
%o from sympy import divisor_count
%o def A183063(n): return divisor_count(n>>(m:=(~n&n-1).bit_length()))*m # _Chai Wah Wu_, Jul 16 2022
%Y Cf. A000005, A000265, A001227, A007814, A000593, A183064, A136655, A125911, A320111.
%Y Column 2 of A195050. - _Omar E. Pol_, Oct 19 2011
%Y Cf. A001620, A027750, A083910, A244009.
%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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