A342082 Numbers with an inferior odd divisor > 1.

9, 12, 15, 18, 21, 24, 25, 27, 30, 33, 35, 36, 39, 40, 42, 45, 48, 49, 50, 51, 54, 55, 56, 57, 60, 63, 65, 66, 69, 70, 72, 75, 77, 78, 80, 81, 84, 85, 87, 90, 91, 93, 95, 96, 98, 99, 100, 102, 105, 108, 110, 111, 112, 114, 115, 117, 119, 120, 121, 123, 125

Offset: 1

Author

Gus Wiseman, Mar 06 2021

Keywords

• nonn

Comments

We define a divisor d|n to be inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.

Numbers n with an odd prime factor <= sqrt(n). - Chai Wah Wu, Mar 09 2021

Examples

			The divisors > 1 of 72 are {2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}, of which {3, 9} are odd and {2, 3, 4, 6, 8} are inferior, with intersection {3}, so 72 is in the sequence.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

The strictly inferior version is the same with A001248 removed.

Positions of terms > 1 in A069288.

The superior version is A116882, with complement A116883.

The complement is A342081.

A006530 selects the greatest prime factor.

A020639 selects the smallest prime factor.

A038548 counts superior (or inferior) divisors, with strict case A056924.

- Odd -

A000009 counts partitions into odd parts, ranked by A066208.

A001227 counts odd divisors.

A026424 lists numbers with odd Omega.

A027193 counts odd-length partitions.

A058695 counts partitions of odd numbers.

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

A340101 counts factorizations into odd factors; A340102 also has odd length.

A340854/A340855 cannot/can be factored with odd minimum factor.

A341594 counts strictly superior odd divisors

A341675 counts superior odd divisors.

- Inferior: A033676, A066839, A161906.

- Superior: A033677, A051283, A059172, A063538/A063539.

- Strictly Inferior A333805, A341674.

- Strictly Superior: A064052/A048098, A341645/A341646.

Cf. A000005, A000203, A001055, A001221, A001222, A001414, A207375, A244991, A300272, A340832, A340931.

Programs

Mathematica

Select[Range[100],Function[n,Select[Divisors[n]//Rest,OddQ[#]&&#<=n/#&]!={}]]

PARI

is(n) = #select(x -> x > 2 && x^2 <= n, factor(n)[, 1]) > 0; \\ Amiram Eldar, Nov 01 2024

Python

from sympy import primefactors
A342082_list = [n for n in range(1,10**3) if len([p for p in primefactors(n) if p > 2 and p*p <= n]) > 0] # Chai Wah Wu, Mar 09 2021