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

1,2

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

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.

Formula

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

Example

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

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 *)

Prog

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

Crossrefs

Contains A000079 and A036896.

Cf. A001227, A033950.

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

Sequence in context: A324524 A325621 A025611 * A251642 A079931 A188915

Adjacent sequences: A049436 A049437 A049438 * A049440 A049441 A049442

Keyword

nice,nonn

Author

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

Extensions

Example corrected by Harvey P. Dale, Jul 14 2011

Status

approved