cp's OEIS Frontend

A341593 Number of superior prime-power divisors of n.

%I A341593 #21 Nov 01 2024 05:15:47
%S A341593 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 A341593 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 A341593 1,0,1,1,1,1,1,1,1,1,1,1,3,1,1,0,1,1,1
%N A341593 Number of superior prime-power divisors of n.
%C A341593 We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.
%H A341593 Amiram Eldar, <a href="/A341593/b341593.txt">Table of n, a(n) for n = 1..10000</a>
%e A341593 The superior prime-power divisors (columns) of selected n:
%e A341593 n = 4374  5103  6144  7500  9000
%e A341593     ----------------------------
%e A341593       81    81   128   125   125
%e A341593      243   243   256   625
%e A341593      729   729   512
%e A341593     2187        1024
%e A341593                 2048
%t A341593 Table[Length[Select[Divisors[n],PrimePowerQ[#]&&#>=n/#&]],{n,100}]
%o A341593 (PARI) a(n) = sumdiv(n, d, d^2 >= n && isprimepower(d)); \\ _Amiram Eldar_, Nov 01 2024
%Y A341593 Positions of zeros after the first are A051283.
%Y A341593 The inferior version is A333750.
%Y A341593 The version for prime instead of prime-power divisors is A341591.
%Y A341593 The version for squarefree instead of prime-power divisors is A341592.
%Y A341593 Dominates A341644 (the strictly superior case).
%Y A341593 The version for odd instead of prime-power divisors is A341675.
%Y A341593 The strictly inferior version is A341677.
%Y A341593 A000961 lists prime powers.
%Y A341593 A001221 counts prime divisors, with sum A001414.
%Y A341593 A001222 counts prime-power divisors.
%Y A341593 A005117 lists squarefree numbers.
%Y A341593 A033677 selects the smallest superior divisor.
%Y A341593 A038548 counts superior (or inferior) divisors.
%Y A341593 A056924 counts strictly superior (or strictly inferior) divisors.
%Y A341593 A161908 lists superior divisors.
%Y A341593 A207375 lists central divisors.
%Y A341593 - Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
%Y A341593 - Superior: A059172, A063538, A063539, A070038, A116882, A116883, A341676.
%Y A341593 - Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.
%Y A341593 - Strictly Superior: A048098, A064052 A140271, A238535, A341594, A341595, A341642, A341643, A341645, A341646, A341673.
%Y A341593 Cf. A000005, A000203, A001248, A006530, A020639, A112798.
%K A341593 nonn
%O A341593 1,4
%A A341593 _Gus Wiseman_, Feb 19 2021