OFFSET
1,1
COMMENTS
For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~7.95n. It follows that the density of the infinitary abundant numbers is 1/7.95, which is about 0.126.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Graeme L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
Eric Weisstein's World of Mathematics, Infinitary Divisor.
EXAMPLE
The third integer that is exceeded by its proper infinitary divisor sum is 40. Hence a(3)=40.
MATHEMATICA
ExponentList[n_Integer, factors_List]:={#, IntegerExponent[n, # ]}&/@factors; InfinitaryDivisors[1]:={1}; InfinitaryDivisors[n_Integer?Positive]:=Module[ { factors=First/@FactorInteger[n], d=Divisors[n] }, d[[Flatten[Position[ Transpose[ Thread[Function[{f, g}, BitOr[f, g]==g][ #, Last[ # ]]]&/@ Transpose[Last/@ExponentList[ #, factors]&/@d]], _?(And@@#&), {1}]] ]] ] Null; properinfinitarydivisorsum[k_]:=Plus@@InfinitaryDivisors[k]-k; InfinitaryAbundantNumberQ[k_]:=If[properinfinitarydivisorsum[k]>k, True, False]; Select[Range[500], InfinitaryAbundantNumberQ[ # ] &]
fun[p_, e_] := Module[{ b = IntegerDigits[e, 2]}, m=Length[b]; Product[If[b[[j]] > 0, 1+p^(2^(m-j)), 1], {j, 1, m}]]; isigma[1]=1; isigma[n_] := Times @@ fun @@@ FactorInteger[n]; Select[Range[1000], isigma[#]>2# &] (* Amiram Eldar, May 12 2019 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Ant King, Apr 29 2007
STATUS
approved