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Number of divisors d of n such that d<e<2*d for at least another divisor e of n.
5

%I #11 Apr 13 2024 05:12:53

%S 0,0,0,0,0,1,0,0,0,0,0,3,0,0,1,0,0,2,0,1,0,0,0,5,0,0,0,1,0,5,0,0,0,0,

%T 1,6,0,0,0,3,0,3,0,0,3,0,0,7,0,0,0,0,0,3,0,3,0,0,0,9,0,0,1,0,0,3,0,0,

%U 0,3,0,9,0,0,2,0,1,2,0,5,0,0,0,9,0,0,0,1,0,9,1,0,0,0,0,9,0,0,1,2,0,2,0,1,4

%N Number of divisors d of n such that d<e<2*d for at least another divisor e of n.

%H Reinhard Zumkeller, <a href="/A174903/b174903.txt">Table of n, a(n) for n = 1..10000</a>

%F a(A174905(n)) = 0; a(A005279(n)) > 0.

%F a(A174904(n)) = n and a(m) <> n for m < A174904(n).

%F a(m)*a(n) <= a(m*n) for m, n coprime.

%e a(12) = #{(2,3), (3,4), (4,6)} = 3;

%e a(15) = #{(3,5)} = 1;

%e a(18) = #{(2,3), (6,9)} = 2;

%e a(20) = #{(4,5)} = 1;

%e a(24) = #{(2,3), (3,4), (4,6), (6,8), (8,12)} = 5.

%t a[n_] := Module[{d = Divisors[n]}, Count[d, _?(Length[Intersection[Range[# + 1, 2*# - 1], d]] > 0 &)]]; Array[a, 100] (* _Amiram Eldar_, Apr 13 2024 *)

%o (Haskell)

%o import Data.List (intersect)

%o a174903 n = length [d | let ds = a027750_row n, d <- ds,

%o not $ null [e | e <- [d+1 .. 2*d-1] `intersect` ds]]

%o -- _Reinhard Zumkeller_, Sep 29 2014

%Y Cf. A000005, A005279, A174904, A174905.

%K nonn

%O 1,12

%A _Reinhard Zumkeller_, Apr 01 2010