OFFSET
1,1
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
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
EXAMPLE
60 has three strictly superior squarefree divisors {10,15,30} so 60 is in the sequence.
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/#&]!={}]]
PROG
(PARI) is(n) = fordiv(n, d, if(d^2 > n && issquarefree(d), return(1))); 0; \\ Amiram Eldar, Nov 01 2024
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.
- Superior: A033677, A063538, A063539, A070038, A072500, A116882, A161908, A341591, A341592, A341593, A341675, A341676.
Subsequence of A007916.
KEYWORD
nonn,changed
AUTHOR
Gus Wiseman, Feb 22 2021
STATUS
approved