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A322287
The number of odd abundant numbers below 10^n.
3
0, 0, 1, 23, 210, 1996, 20661, 205366, 2048662, 20502004, 204951472
OFFSET
1,4
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.
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.
FORMULA
Lim_{n->oo} a(n)/10^n = 0.0020... is the density of odd abundant numbers.
EXAMPLE
945 is the only odd abundant number below 10^3, thus a(3) = 1.
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
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Aug 28 2019
STATUS
approved