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%I #7 Feb 23 2021 08:30:38
%S 0,0,0,0,0,1,0,1,0,1,0,2,0,1,1,1,0,2,0,2,1,1,0,3,0,1,1,2,0,3,0,2,1,1,
%T 1,3,0,1,1,3,0,2,0,2,2,1,0,3,0,2,1,2,0,2,1,3,1,1,0,4,0,1,2,2,1,2,0,2,
%U 1,3,0,4,0,1,2,2,1,2,0,4,1,1,0,4,1,1,1
%N Number of strictly inferior prime-power 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.
%e The strictly inferior prime-power divisors of n!:
%e n = 1 2 6 24 120 720 5040 40320
%e ----------------------------------
%e . . 2 2 2 2 2 2
%e 3 3 3 3 3
%e 4 4 4 4 4
%e 5 5 5 5
%e 8 8 7 7
%e 9 8 8
%e 16 9 9
%e 16 16
%e 32
%e 64
%e 128
%t Table[Length[Select[Divisors[n],PrimePowerQ[#]&&#<n/#&]],{n,100}]
%Y Positions of zeros are A166684.
%Y The weakly inferior version is A333750.
%Y The version for odd instead of prime-power divisors is A333805.
%Y The version for prime instead of prime-power divisors is A333806.
%Y The weakly superior version is A341593.
%Y The version for squarefree instead of prime-power divisors is A341596.
%Y The strictly superior version is A341644.
%Y A000961 lists prime powers.
%Y A001221 counts prime divisors, with sum A001414.
%Y A001222 counts prime-power divisors.
%Y A005117 lists squarefree numbers.
%Y A038548 counts superior (or inferior) divisors.
%Y A056924 counts strictly superior (or strictly inferior) divisors.
%Y A207375 lists central divisors.
%Y - Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
%Y - Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A161908, A341591, A341592, A341676.
%Y - Strictly Inferior: A060775, A070039, A341674.
%Y - Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341645, A341646, A341673.
%Y Cf. A000005, A000203, A000430, A001248, A006530, A020639.
%K nonn
%O 1,12
%A _Gus Wiseman_, Feb 23 2021
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