%I #45 Jun 12 2022 02:58:56

%S 1,2,4,8,9,16,18,32,36,64,72,128,144,225,256,288,441,450,512,576,625,

%T 882,900,1024,1089,1152,1250,1521,1764,1800,2025,2048,2178,2304,2500,

%U 2601,3042,3249,3528,3600,4050,4096,4356,4608,4761,5000,5202,5625,6084

%N Numbers k such that the number of odd divisors of k is an odd divisor of k.

%C Invented by the HR concept formation program.

%C Sequence consists of all numbers of the form A000079(k)*A036896(m). - _Matthew Vandermast_, Nov 14 2010

%H Reinhard Zumkeller, <a href="/A049439/b049439.txt">Table of n, a(n) for n = 1..1000</a>

%H Simon Colton, <a href="http://www.cs.uwaterloo.ca/journals/JIS/colton/joisol.html">Refactorable Numbers - A Machine Invention</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.

%H Simon Colton, <a href="http://web.archive.org/web/20070831060523/http://www.dai.ed.ac.uk/homes/simonco/research/hr/">HR - Automatic Theory Formation in Pure Mathematics</a>.

%F a(n) = A000079(k)*A016754(m) for appropriate k, m. - _Reinhard Zumkeller_, Jun 05 2008

%e There are 3 odd divisors of 18, namely 1,3 and 9 and 3 itself is an odd divisor of 18.

%t ok[n_] := (d = Length @ Select[Divisors[n], OddQ] ;

%t IntegerQ[n/d] && OddQ[d]); Select[Range[6100], ok]

%t (* _Jean-François Alcover_, Apr 22 2011 *)

%t odQ[n_]:=Module[{ods=Select[Divisors[n],OddQ]},MemberQ[ods,Length[ ods]]]; Select[Range[7000],odQ] (* _Harvey P. Dale_, Dec 18 2011 *)

%t Select[Range[6000], OddQ[(d = DivisorSigma[0, #/2^IntegerExponent[#, 2]])] && Divisible[#, d] &] (* _Amiram Eldar_, Jun 12 2022 *)

%o (Haskell)

%o a049439 n = a049439_list !! (n-1)

%o a049439_list = filter (\x -> ((length $ oddDivs x) `elem` oddDivs x)) [1..]

%o where oddDivs n = [d | d <- [1,3..n], mod n d == 0]

%o -- _Reinhard Zumkeller_, Aug 17 2011

%o (PARI) is(n)=my(d=numdiv(n>>valuation(n,2))); d%2 && n%d==0 \\ _Charles R Greathouse IV_, Feb 07 2017

%Y Contains A000079 and A036896.

%Y Cf. A001227, A033950.

%Y Subsequence of A028982. Includes A120349, A120358, A120359, A120361, A181795. See also A181794.

%K nice,nonn

%O 1,2

%A Simon Colton (simonco(AT)cs.york.ac.uk)

%E Example corrected by _Harvey P. Dale_, Jul 14 2011