A341646 Numbers with a strictly superior squarefree divisor.

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Author

Gus Wiseman, Feb 22 2021

Keywords

• nonn

Comments

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

This is a subsequence of A007916, i.e., no perfect powers appear here. [For perfect powers n, a supposed strictly superior squarefree divisor d=p*q*r... with distinct primes p,q,r,s... has a complementary divisor n/d=p^i*q^j*r^k*s*... with i,j,k>=1, so the complementary divisor is at least as large as d, a contradiction.] Entries in A007916 but not in here are 48, 54, 72, 96, 108, 160, 162, 192,... - R. J. Mathar, Jul 07 2023

Is this a duplicate of A089105? - R. J. Mathar, Jul 24 2023

Examples

			60 has three strictly superior squarefree divisors {10,15,30} so 60 is in the sequence.
		

Links

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

Crossrefs

The version for prime instead of squarefree divisors is A064052.

The version for prime-power instead of squarefree divisors is the complement of A051283.

The weakly superior version is the complement of A059172.

The version for odd instead of squarefree divisors is A116883.

These are the positions of nonzero terms in A341595.

The complement is A341645.

A005117 lists squarefree numbers.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A140271 selects the smallest strictly superior divisor.

A207375 list central divisors.

A341673 lists strictly superior divisors.

- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.

- Superior: A033677, A063538, A063539, A070038, A072500, A116882, A161908, A341591, A341592, A341593, A341675, A341676.

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

- Strictly Superior: A048098, A238535, A341594, A341595, A341643, A341644.

Cf. A001222, A001248, A006530, A020639, A112798.

Subsequence of A007916.

Programs

Maple

isA341646 := proc(n)
    local d ;
    for d in numtheory[divisors](n) do
        if d>n/d then
            if issqrfree(d) then
                return true ;
            end if;
        end if;
    end do:
    false ;
end proc:
for n from 2 to 100 do
    if isA341646(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 07 2023

Mathematica

Select[Range[100],Function[n,Select[Divisors[n],SquareFreeQ[#]&&#>n/#&]!={}]]

PARI

is(n) = fordiv(n, d, if(d^2 > n && issquarefree(d), return(1))); 0; \\ Amiram Eldar, Nov 01 2024