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%I #23 Sep 09 2022 04:19:39
%S 945,10395,12285,15015,16065,17955,19305,19635,21735,21945,23205,
%T 23625,25245,25935,26565,27405,28215,28875,29295,29835,31395,33345,
%U 33495,33915,34125,34155,34965,35805,37125,38745,39585,40635,41055,42315
%N Odd infinitary abundant numbers.
%C This is also the sequence of odd integers whose infinitary aliquot sequences initially increase. Based on empirical evidence (up to 10 million), this applies to only about 0.1% of odd integers.
%C The numbers of terms not exceeding 10^k, for k = 4, 5, ..., are 1, 77, 473, 5703, 53569, 561610, 5525461, 54979537, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0005... . - _Amiram Eldar_, Sep 09 2022
%H Amiram Eldar, <a href="/A127666/b127666.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., Vol. 54, No. 189, (1990), 395-411.
%H J. O. M. Pedersen, <a href="http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Broken link]
%H J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a> [Via Internet Archive Wayback-Machine]
%H J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a> [Cached copy, pdf file only]
%F Odd values of n for which A126168(n)>n.
%e a(5)=16065 because 16065 is the fifth odd number that is exceeded by the sum of its proper infinitary divisors.
%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;Select[Range[1,50000,2],properinfinitarydivisorsum[ # ]># &] (* end of program *)
%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[1, 50000, 2], isigma[#] > 2 # &] (* _Amiram Eldar_, Jun 09 2019 *)
%o (PARI) A049417(n) = {my(b, f=factorint(n)); prod(k=1, #f[, 2], b = binary(f[k, 2]); prod(j=1, #b, if(b[j], 1+f[k, 1]^(2^(#b-j)), 1)))}
%o isok(k) = A049417(k)>2*k&&k%2==1; \\ _Jinyuan Wang_, Jun 09 2019
%Y Cf. A005231, A126168, A127661, A129656.
%K nonn
%O 1,1
%A _Ant King_, Jan 26 2007
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