%I #21 Nov 01 2024 05:15:47

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

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

%U 1,0,1,1,1,1,1,1,1,1,1,1,3,1,1,0,1,1,1

%N Number of superior prime-power 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="/A341593/b341593.txt">Table of n, a(n) for n = 1..10000</a>

%e The superior prime-power divisors (columns) of selected n:

%e n = 4374 5103 6144 7500 9000

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

%e 81 81 128 125 125

%e 243 243 256 625

%e 729 729 512

%e 2187 1024

%e 2048

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

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

%Y Positions of zeros after the first are A051283.

%Y The inferior version is A333750.

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

%Y The version for squarefree instead of prime-power divisors is A341592.

%Y Dominates A341644 (the strictly superior case).

%Y The version for odd instead of prime-power divisors is A341675.

%Y The strictly inferior version is A341677.

%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 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, A333749.

%Y - Superior: A059172, A063538, A063539, A070038, A116882, A116883, A341676.

%Y - Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.

%Y - Strictly Superior: A048098, A064052 A140271, A238535, A341594, A341595, A341642, A341643, A341645, A341646, A341673.

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

%K nonn

%O 1,4

%A _Gus Wiseman_, Feb 19 2021