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A341646 Numbers with a strictly superior squarefree divisor.

%I A341646 #24 Nov 01 2024 05:14:44
%S A341646 2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,28,29,30,31,
%T A341646 33,34,35,37,38,39,40,41,42,43,44,45,46,47,50,51,52,53,55,56,57,58,59,
%U A341646 60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80
%N A341646 Numbers with a strictly superior squarefree divisor.
%C A341646 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.
%C A341646 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
%C A341646 Is this a duplicate of A089105? - _R. J. Mathar_, Jul 24 2023
%H A341646 Amiram Eldar, <a href="/A341646/b341646.txt">Table of n, a(n) for n = 1..10000</a>
%e A341646 60 has three strictly superior squarefree divisors {10,15,30} so 60 is in the sequence.
%p A341646 isA341646 := proc(n)
%p A341646     local d ;
%p A341646     for d in numtheory[divisors](n) do
%p A341646         if d>n/d then
%p A341646             if issqrfree(d) then
%p A341646                 return true ;
%p A341646             end if;
%p A341646         end if;
%p A341646     end do:
%p A341646     false ;
%p A341646 end proc:
%p A341646 for n from 2 to 100 do
%p A341646     if isA341646(n) then
%p A341646         printf("%d,",n) ;
%p A341646     end if;
%p A341646 end do: # _R. J. Mathar_, Jul 07 2023
%t A341646 Select[Range[100],Function[n,Select[Divisors[n],SquareFreeQ[#]&&#>n/#&]!={}]]
%o A341646 (PARI) is(n) = fordiv(n, d, if(d^2 > n && issquarefree(d), return(1))); 0; \\ _Amiram Eldar_, Nov 01 2024
%Y A341646 The version for prime instead of squarefree divisors is A064052.
%Y A341646 The version for prime-power instead of squarefree divisors is the complement of A051283.
%Y A341646 The weakly superior version is the complement of A059172.
%Y A341646 The version for odd instead of squarefree divisors is A116883.
%Y A341646 These are the positions of nonzero terms in A341595.
%Y A341646 The complement is A341645.
%Y A341646 A005117 lists squarefree numbers.
%Y A341646 A038548 counts superior (or inferior) divisors.
%Y A341646 A056924 counts strictly superior (or strictly inferior) divisors.
%Y A341646 A140271 selects the smallest strictly superior divisor.
%Y A341646 A207375 list central divisors.
%Y A341646 A341673 lists strictly superior divisors.
%Y A341646 - Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
%Y A341646 - Superior: A033677, A063538, A063539, A070038, A072500, A116882, A161908, A341591, A341592, A341593, A341675, A341676.
%Y A341646 - Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.
%Y A341646 - Strictly Superior: A048098, A238535, A341594, A341595, A341643, A341644.
%Y A341646 Cf. A001222, A001248, A006530, A020639, A112798.
%Y A341646 Subsequence of A007916.
%K A341646 nonn
%O A341646 1,1
%A A341646 _Gus Wiseman_, Feb 22 2021