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Number of strictly superior prime divisors of n.
27

%I #11 Nov 01 2024 05:15:59

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

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

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

%N Number of strictly superior prime divisors of n.

%C We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

%C All terms are binary numbers.

%H Amiram Eldar, <a href="/A341642/b341642.txt">Table of n, a(n) for n = 1..10000</a>

%e The sequence of sets of strictly superior prime divisors of each positive integer begins: {}, {2}, {3}, {}, {5}, {3}, {7}, {}, {}, {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 zeros are A048098.

%Y The inferior version is A063962.

%Y Positions of ones are A064052.

%Y The strictly inferior version is A333806.

%Y Dominated by A341591 (the weak version).

%Y The version for odd instead of prime divisors is A341594.

%Y The version for squarefree instead of prime divisors is A341595.

%Y The strictly superior prime divisors themselves are listed by A341643.

%Y The version for prime-power instead of prime divisors is A341644.

%Y A001221 counts prime divisors, with sum A001414.

%Y A140271 selects the smallest strictly superior divisor.

%Y A038548 counts superior (or inferior) divisors.

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

%Y A207375 list central divisors.

%Y A341673 lists strictly superior divisors.

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

%Y - Superior: A033677, A051283, A059172, A063538, A063539, A070038, A072500, A116882, A116883, A161908, A341592, A341593, A341675, A341676.

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

%Y - Strictly Superior: A238535, A341644, A341646.

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

%K nonn

%O 1

%A _Gus Wiseman_, Feb 21 2021