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A322287 The number of odd abundant numbers below 10^n. 3

%I #11 Sep 02 2019 08:12:42

%S 0,0,1,23,210,1996,20661,205366,2048662,20502004,204951472

%N The number of odd abundant numbers below 10^n.

%C Anderson proved that the density of odd deficient numbers is at least (48 - 3*Pi^2)/(32 - Pi^2) ~ 0.831...

%C Kobayashi et al. proved that the density of odd abundant numbers is between 0.002042 and 0.002071.

%H C. W. Anderson, <a href="https://www.jstor.org/stable/2318269">Density of Deficient Odd Numbers</a>, The American Mathematical Monthly, Vol. 82, No. 10 (1975), pp. 1018-1020.

%H Mitsuo Kobayashi, Paul Pollack and Carl Pomerance, <a href="https://doi.org/10.1016/j.jnt.2008.10.011">On the distribution of sociable numbers</a>, Journal of Number Theory, Vol. 129, No. 8 (2009), pp. 1990-2009. See Theorem 10 on p. 2007.

%F Lim_{n->oo} a(n)/10^n = 0.0020... is the density of odd abundant numbers.

%e 945 is the only odd abundant number below 10^3, thus a(3) = 1.

%t abQ[n_] := DivisorSigma[1, n] > 2 n; c = 0; k = 1; s = {}; Do[While[k < 10^n, If[abQ[k], c++]; k += 2]; AppendTo[s, c], {n, 1, 5}]; s

%Y Cf. A000203, A005231, A302992, A302993, A302994, A307820, A307821, A307823.

%K nonn,more

%O 1,4

%A _Amiram Eldar_, Aug 28 2019

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