A069288 Number of odd divisors of n <= sqrt(n).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 2, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 3, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 3, 1, 1, 3, 1, 2, 2, 1, 1, 2, 3, 1, 2, 1, 1, 3, 1, 2, 2, 1, 2, 3, 1, 1, 3, 2, 1, 2, 1, 1, 4

Offset: 1

Author

Reinhard Zumkeller, Mar 14 2002

Keywords

• nonn

Comments

a(n) = #{d : d = A182469(n,k), d <= A000196(n), k=1..A001227(n)}. - Reinhard Zumkeller, Apr 05 2015

Examples

			From _Gus Wiseman_, Feb 11 2021: (Start)
The inferior odd divisors for selected n are the columns below:
n: 1    9   30   90  225  315  630  945 1575 2835 4410 3465 8190 6930
  --------------------------------------------------------------------
   1    3    5    9   15   15   21   27   35   45   63   55   65   77
        1    3    5    9    9   15   21   25   35   49   45   63   63
             1    3    5    7    9   15   21   27   45   35   45   55
                  1    3    5    7    9   15   21   35   33   39   45
                       1    3    5    7    9   15   21   21   35   35
                            1    3    5    7    9   15   15   21   33
                                 1    3    5    7    9   11   15   21
                                      1    3    5    7    9   13   15
                                           1    3    5    7    9   11
                                                1    3    5    7    9
                                                     1    3    5    7
                                                          1    3    5
                                                               1    3
                                                                    1
(End)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000196, A001227, A069289, A182469.

Positions of first appearances are A334853.

A055396 selects the least prime index.

A061395 selects the greatest prime index.

- Odd -

A000009 counts partitions into odd parts (A066208).

A026424 lists numbers with odd Omega.

A027193 counts odd-length partitions.

A067659 counts strict partitions of odd length (A030059).

- Inferior divisors -

A033676 selects the greatest inferior divisor.

A033677 selects the least superior divisor.

A038548 counts inferior divisors.

A060775 selects the greatest strictly inferior divisor.

A063538 lists numbers with a superior prime divisor.

A063539 lists numbers without a superior prime divisor.

A063962 counts inferior prime divisors.

A064052 lists numbers with a properly superior prime divisor.

A140271 selects the least properly superior divisor.

A217581 selects the greatest inferior divisor.

A333806 counts strictly inferior prime divisors.

Cf. A001055, A244991, A300272, A340101, A340607, A340832, A340854/A340855.

Programs

Haskell

a069288 n = length $ takeWhile (<= a000196 n) $ a182469_row n
-- Reinhard Zumkeller, Apr 05 2015

Mathematica

odn[n_]:=Count[Divisors[n],?(OddQ[#]&&#<=Sqrt[n ]&)]; Array[odn,100] (* _Harvey P. Dale, Nov 04 2017 *)

PARI

a(n) = my(ir = sqrtint(n)); sumdiv(n, d, (d % 2) * (d <= ir)); \\ Michel Marcus, Jan 14 2014

Formula

G.f.: Sum_{n>=1} 1/(1-q^(2*n-1)) * q^((2*n-1)^2). [Joerg Arndt, Mar 04 2010]