A341675 Number of superior odd divisors of n.

1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 1, 1, 1, 0, 2, 1, 2, 1, 1, 1, 2, 0, 1, 2, 1, 1, 3, 1, 1, 0, 2, 1, 2, 1, 1, 2, 2, 0, 2, 1, 1, 1, 1, 1, 3, 0, 2, 2, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 2, 2, 1, 0, 3, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 0, 1, 1, 3, 1, 1, 2, 1, 1, 4

Offset: 1

Author

Gus Wiseman, Feb 23 2021

Keywords

• nonn

Comments

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.

Examples

			The superior odd divisors of selected n:
n = 1  9  45  105  225  315  945  1575  1890  2835  3465  8190
   -----------------------------------------------------------
    1  9  45  105  225  315  945  1575  945   2835  3465  4095
       3  15  35   75   105  315  525   315   945   1155  1365
          9   21   45   63   189  315   189   567   693   819
              15   25   45   135  225   135   405   495   585
                   15   35   105  175   105   315   385   455
                        21   63   105   63    189   315   315
                             45   75    45    135   231   273
                             35   63          105   165   195
                                  45          81    105   117
                                              63    99    105
                                                    77    91
                                                    63
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

The inferior version is A069288.

Positions of 0's (except a(1)) are A116882.

The version for prime instead of odd divisors is A341591.

The version for squarefree instead of odd divisors is A341592.

The version for prime-power instead of odd divisors is A341593.

The strictly superior case is A341594.

A001221 counts prime divisors, with sum A001414.

A005117 lists squarefree numbers.

A033676 selects the greatest inferior divisor.

A033677 selects the smallest superior divisor.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A161908 lists superior divisors.

A207375 lists central divisors.

- Inferior: A063962, A066839, A161906, A217581, A333749, A333750.

- Superior: A051283, A059172, A063538, A063539, A070038, A116882, A341676.

- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.

- Strictly Superior: A048098, A064052 A140271, A238535, A341595, A341642, A341643, A341644, A341645, A341646, A341673.

Cf. A000005, A000203, A001222, A001248, A006530, A020639.

Programs

Mathematica

Table[Length[Select[Divisors[n],OddQ[#]&&#>=n/#&]],{n,100}]

PARI

A341675(n) = sumdiv(n,d,(d%2)*(d>=n/d)); \\ Antti Karttunen, Dec 13 2021

Extensions

Data section extended up to 105 terms by Antti Karttunen, Dec 13 2021