A333805 Number of odd divisors of n that are < sqrt(n).

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

Offset

1,12

Comments

If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence counts strictly inferior odd divisors. - Gus Wiseman, Feb 26 2021

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

G.f.: Sum_{k>=1} x^(2*k*(2*k - 1)) / (1 - x^(2*k - 1)).

Example

The strictly inferior odd divisors of 945 are 1, 3, 5, 7, 9, 15, 21, 27, so a(945) = 8. - Gus Wiseman, Feb 27 2021

Mathematica

Table[DivisorSum[n, 1 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Prog

(PARI) A333805(n) = sumdiv(n, d, (d%2)&&((d*d)<n)); \\ Antti Karttunen, Nov 02 2022

Crossrefs

Dominated by A001227 (number of odd divisors).

Strictly inferior divisors (not just odd) are counted by A056924.

The non-strict version is A069288.

These divisors add up to A070039.

The case of prime divisors is A333806.

The strictly superior version is A341594.

The case of squarefree divisors is A341596.

The superior version is A341675.

The case of prime-power divisors is A341677.

A006530 selects the greatest prime factor.

A020639 selects the smallest prime factor.

- Odd -

A000009 counts partitions into odd parts, ranked by A066208.

A026424 lists numbers with odd Omega.

A027193 counts odd-length partitions.

A067659 counts strict partitions of odd length, ranked by A030059.

- Inferior divisors -

A033676 selects the greatest inferior divisor.

A033677 selects the smallest superior divisor.

A038548 counts superior (or inferior) divisors.

A060775 selects the greatest strictly inferior divisor.

A341674 lists strictly inferior divisors.

Cf. A001248, A051283, A063538/A063539, A063962, A116883/A116882, A300272, A333807, A333809, A340832.

Sequence in context: A152846 A152845 A115571 * A152844 A115580 A115583

Adjacent sequences: A333802 A333803 A333804 * A333806 A333807 A333808

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 05 2020

Extensions

Data section extended up to a(105) by Antti Karttunen, Nov 02 2022

Status

approved