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Number of squarefree superior divisors of n.
26

%I #18 Nov 01 2024 05:15:27

%S 1,1,1,1,1,2,1,0,1,2,1,1,1,2,2,0,1,1,1,2,2,2,1,1,1,2,0,2,1,4,1,0,2,2,

%T 2,1,1,2,2,1,1,4,1,2,1,2,1,0,1,1,2,2,1,0,2,1,2,2,1,3,1,2,1,0,2,4,1,2,

%U 2,4,1,0,1,2,1,2,2,4,1,1,0,2,1,3,2,2,2

%N Number of squarefree superior divisors of n.

%C We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.

%H Amiram Eldar, <a href="/A341592/b341592.txt">Table of n, a(n) for n = 1..10000</a>

%e The strictly superior squarefree divisors (columns) of selected n:

%e 1 6 8 30 60 210 420 630 1050 2310 4620 6930

%e ------------------------------------------------------

%e 1 3 . 6 10 15 21 30 35 55 70 105

%e 6 10 15 21 30 35 42 66 77 110

%e 15 30 30 35 42 70 70 105 154

%e 30 35 42 70 105 77 110 165

%e 42 70 105 210 105 154 210

%e 70 105 210 110 165 231

%e 105 210 154 210 330

%e 210 165 231 385

%e 210 330 462

%e 231 385 770

%e 330 462 1155

%e 385 770 2310

%e 462 1155

%e 770 2310

%e 1155

%e 2310

%p with(numtheory):

%p a := n -> nops(select(d -> d*d >= n and issqrfree(d), divisors(n))):

%p seq(a(n), n = 1..88); # _Peter Luschny_, Feb 20 2021

%t Table[Length[Select[Divisors[n],SquareFreeQ[#]&&#>=n/#&]],{n,100}]

%o (PARI) a(n) = sumdiv(n, d, d^2 >= n && issquarefree(d)); \\ _Amiram Eldar_, Nov 01 2024

%Y Positions of zeros are A059172.

%Y The inferior version is A333749.

%Y The version for prime instead of squarefree divisors is A341591.

%Y The version for prime powers instead of squarefree divisors is A341593.

%Y The strictly superior case is A341595.

%Y The version for odd instead of squarefree divisors is A341675.

%Y A001221 counts prime divisors, with sum A001414.

%Y A033677 selects the smallest superior divisor.

%Y A038548 counts superior (or inferior) divisors.

%Y A056924 counts strictly superior (or strictly inferior) divisors.

%Y A161908 lists superior divisors.

%Y A207375 lists central divisors.

%Y - Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333750.

%Y - Superior: A051283, A063538, A063539, A070038, A116882, A116883, A341676.

%Y - Strictly Inferior: A060775, A333805, A333806, A341596, A341674.

%Y - Strictly Superior: A048098, A064052 A140271, A238535, A341594, A341642, A341643, A341644, A341645, A341646, A341673.

%Y Cf. A000005, A000203, A001222, A001248, A006530, A020639, A112798.

%K nonn

%O 1,6

%A _Gus Wiseman_, Feb 19 2021