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

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

Offset: 1

Author

Amiram Eldar, Aug 28 2019

Keywords

• nonn
• more

Comments

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

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

Examples

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

Links

• C. W. Anderson, Density of Deficient Odd Numbers, The American Mathematical Monthly, Vol. 82, No. 10 (1975), pp. 1018-1020.
• Mitsuo Kobayashi, Paul Pollack and Carl Pomerance, On the distribution of sociable numbers, Journal of Number Theory, Vol. 129, No. 8 (2009), pp. 1990-2009. See Theorem 10 on p. 2007.

Crossrefs

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

Programs

Mathematica

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

Formula

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