A181794 -id:A181794 - cp's OEIS frontend

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

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

Invented by the HR concept formation program.

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

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

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.

Contains A000079 and A036896.
Cf. A001227, A033950.
Subsequence of A028982. Includes A120349, A120358, A120359, A120361, A181795. See also A181794.

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

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 *)

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

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

Example corrected by Harvey P. Dale, Jul 14 2011

A181795 Numbers k such that the number of odd divisors of k is an odd divisor of k, and the number of even divisors of k is an even divisor of k.

Original entry on oeis.org

4, 16, 36, 144, 256, 576, 900, 1764, 2304, 2500, 3600, 4356, 6084, 7056, 8100, 10000, 10404, 12996, 17424, 19044, 22500, 24336, 26244, 30276, 32400, 34596, 36864, 41616, 49284, 51984, 57600, 60516, 65536, 66564, 76176, 79524, 90000

Offset: 1

Matthew Vandermast, Nov 14 2010

nonn

All members are even squares (A016742). Intersection of A049439 and A181794.

Includes all numbers of the form A001146(m)*A036896(n) for m>1.

			a(3)=36 has 3 odd divisors (1, 3, and 9) and 6 even divisors (2, 4, 6, 12, 18, and 36). 3 and 6 are odd and even respectively, and both are divisors of 36.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..96 from Harvey P. Dale)

Subsequence of A000290, A016742, A120351.

See also A033950,A181687. For refactorable members of this sequence, see A120349.

ndQ[n_]:=Module[{d=Divisors[n],od,ev},od=Count[d,?OddQ];ev=Count[ d, ?EvenQ]; ev!=0&&OddQ[od]&&EvenQ[ev]&&Divisible[n,od]&&Divisible[ n, ev]]; Select[Range[100000],ndQ] (* Harvey P. Dale, Feb 24 2016 *)

isok(n) = my(nod = numdiv(n>>valuation(n, 2)), noe = if (n%2, 0, numdiv(n/2))); (nod % 2) && nod && !(n % nod) && !(noe % 2) && noe && !(n % noe); \\ Michel Marcus, Jan 14 2014

More terms from Nathaniel Johnston, Nov 17 2010

cp's OEIS Frontend

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

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