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Number of superior prime divisors of n.
29

%I #23 Nov 01 2024 05:15:37

%S 0,1,1,1,1,1,1,0,1,1,1,0,1,1,1,0,1,0,1,1,1,1,1,0,1,1,0,1,1,0,1,0,1,1,

%T 1,0,1,1,1,0,1,1,1,1,0,1,1,0,1,0,1,1,1,0,1,0,1,1,1,0,1,1,0,0,1,1,1,1,

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

%N Number of superior prime 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.

%C All terms are binary numbers.

%H Amiram Eldar, <a href="/A341591/b341591.txt">Table of n, a(n) for n = 1..10000</a>

%e The sequence of sets of superior prime divisors of each positive integer begins: {}, {2}, {3}, {2}, {5}, {3}, {7}, {}, {3}, {5}, {11}, {}, {13}, {7}, {5}, {}, {17}, {}, {19}, {5}, ...

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

%t a[n_] := Count[FactorInteger[n][[;; , 1]], _?(#^2 >= n &)]; a[1] = 0; Array[a, 100] (* _Amiram Eldar_, Nov 01 2024 *)

%o (PARI) a(n) = #select(x -> (x^2 >= n), factor(n)[, 1]); \\ _Amiram Eldar_, Nov 01 2024

%Y Positions of ones are A063538.

%Y Positions of zeros are A063539.

%Y The inferior version is A063962.

%Y The strictly inferior version is A333806.

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

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

%Y Dominates A341642 (the strictly superior version).

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

%Y The unique superior prime divisors of the positive positions are A341676.

%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 list central divisors.

%Y - Inferior: A033676, A066839, A069288, A161906, A217581, A333749, A333750.

%Y - Superior: A051283, A059172, A070038, A072500, A116882, A116883.

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

%Y - Strictly Superior: A048098, A064052 A140271, A238535, A341594, A341595, A341644, A341645, A341646, A341673.

%Y Cf. A000005, A000203, A001222, A001248, A006530, A020639, A112798, A341643.

%K nonn

%O 1

%A _Gus Wiseman_, Feb 19 2021