A049439 Numbers k such that the number of odd divisors of k is an odd divisor of k.

1, 2, 4, 8, 9, 16, 18, 32, 36, 64, 72, 128, 144, 225, 256, 288, 441, 450, 512, 576, 625, 882, 900, 1024, 1089, 1152, 1250, 1521, 1764, 1800, 2025, 2048, 2178, 2304, 2500, 2601, 3042, 3249, 3528, 3600, 4050, 4096, 4356, 4608, 4761, 5000, 5202, 5625, 6084

Offset: 1

Author

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

Keywords

• nice
• nonn

Comments

Invented by the HR concept formation program.

Sequence consists of all numbers of the form A000079(k)*A036896(m). - Matthew Vandermast, Nov 14 2010

Examples

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
• Simon Colton, Refactorable Numbers - A Machine Invention, J. Integer Sequences, Vol. 2 (1999), Article 99.1.2.
• Simon Colton, HR - Automatic Theory Formation in Pure Mathematics.

Crossrefs

Contains A000079 and A036896.

Cf. A001227, A033950.

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

Programs

Haskell

a049439 n = a049439_list !! (n-1)
a049439_list = filter (\x -> ((length $ oddDivs x) `elem` oddDivs x)) [1..]
   where oddDivs n = [d | d <- [1,3..n], mod n d == 0]
-- Reinhard Zumkeller, Aug 17 2011

Mathematica

ok[n_] := (d = Length @ Select[Divisors[n], OddQ] ;
  IntegerQ[n/d] && OddQ[d]); Select[Range[6100], ok]
(* Jean-François Alcover, Apr 22 2011 *)
odQ[n_]:=Module[{ods=Select[Divisors[n],OddQ]},MemberQ[ods,Length[ ods]]]; Select[Range[7000],odQ] (* Harvey P. Dale, Dec 18 2011 *)
Select[Range[6000], OddQ[(d = DivisorSigma[0, #/2^IntegerExponent[#, 2]])] && Divisible[#, d] &] (* Amiram Eldar, Jun 12 2022 *)

PARI

is(n)=my(d=numdiv(n>>valuation(n,2))); d%2 && n%d==0 \\ Charles R Greathouse IV, Feb 07 2017

Formula

a(n) = A000079(k)*A016754(m) for appropriate k, m. - Reinhard Zumkeller, Jun 05 2008

Extensions

Example corrected by Harvey P. Dale, Jul 14 2011