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Number of strictly inferior squarefree divisors of n.
29

%I #13 Nov 01 2024 05:14:55

%S 0,1,1,1,1,2,1,2,1,2,1,3,1,2,2,2,1,3,1,2,2,2,1,3,1,2,2,2,1,4,1,2,2,2,

%T 2,3,1,2,2,3,1,4,1,2,3,2,1,4,1,3,2,2,1,4,2,3,2,2,1,5,1,2,3,2,2,4,1,2,

%U 2,4,1,4,1,2,3,2,2,4,1,3,2,2,1,5,2,2,2

%N Number of strictly inferior squarefree divisors of n.

%C We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.

%H Amiram Eldar, <a href="/A341596/b341596.txt">Table of n, a(n) for n = 1..10000</a>

%e The strictly inferior squarefree divisors of selected n:

%e n = 1 2 6 12 30 60 120 210 240 420 630 1050 1260

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

%e {} 1 1 1 1 1 1 1 1 1 1 1 1

%e 2 2 2 2 2 2 2 2 2 2 2

%e 3 3 3 3 3 3 3 3 3 3

%e 5 5 5 5 5 5 5 5 5

%e 6 6 6 6 6 6 6 6

%e 10 7 10 7 7 7 7

%e 10 15 10 10 10 10

%e 14 14 14 14 14

%e 15 15 15 15

%e 21 21 21

%e 30 30

%e 35

%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 ones are A000430.

%Y The weakly inferior version is A333749.

%Y The version counting odd instead of squarefree divisors is A333805.

%Y The version counting prime instead of squarefree divisors is A333806.

%Y The weakly superior version is A341592.

%Y The strictly superior version is A341595.

%Y The version counting prime-power instead of squarefree divisors is A341677.

%Y A001221 counts prime divisors, with sum A001414.

%Y A001222 counts prime power divisors.

%Y A005117 lists squarefree numbers.

%Y A033676 selects the greatest inferior divisor.

%Y A033677 selects the smallest superior divisor.

%Y A038548 counts superior (or inferior) divisors.

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

%Y A207375 lists central divisors.

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

%Y - Superior: A051283, A059172, A063538, A063539, A070038, A116882, A116883, A161908, A341591, A341593, A341675, A341676.

%Y - Strictly Inferior: A060775, A070039, A341674.

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

%Y Cf. A000005, A000203, A001055, A001248, A050320.

%K nonn

%O 1,6

%A _Gus Wiseman_, Feb 23 2021