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%I #17 Feb 20 2023 12:26:58
%S 24,30,40,42,54,56,66,70,72,78,88,96,102,104,114,120,138,150,168,174,
%T 186,210,216,222,246,258,264,270,280,282,294,312,318,330,354,360,366,
%U 378,384,390,402,408,420,426,438,440,456,462,474,480,486,498
%N Infinitary abundant numbers: integers for which A126168 (n)>n, or equivalently for which A049417 (n)>2n.
%C 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.
%H Amiram Eldar, <a href="/A129656/b129656.txt">Table of n, a(n) for n = 1..10000</a>
%H Graeme L. Cohen, <a href="http://dx.doi.org/10.1090/S0025-5718-1990-0993927-5">On an integer's infinitary divisors</a>, Math. Comp., 54 (1990), 395-411.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfinitaryDivisor.html">Infinitary Divisor</a>.
%e The third integer that is exceeded by its proper infinitary divisor sum is 40. Hence a(3)=40.
%t 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[ # ] &]
%t 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 *)
%Y Cf. A126168, A049417, A127666, A129657, A007357.
%K easy,nonn
%O 1,1
%A _Ant King_, Apr 29 2007
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