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A127666
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Odd infinitary abundant numbers.
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17
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945, 10395, 12285, 15015, 16065, 17955, 19305, 19635, 21735, 21945, 23205, 23625, 25245, 25935, 26565, 27405, 28215, 28875, 29295, 29835, 31395, 33345, 33495, 33915, 34125, 34155, 34965, 35805, 37125, 38745, 39585, 40635, 41055, 42315
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OFFSET
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1,1
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COMMENTS
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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.
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
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LINKS
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FORMULA
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Odd values of n for which A126168(n)>n.
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EXAMPLE
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a(5)=16065 because 16065 is the fifth odd number that is exceeded by the sum of its proper infinitary divisors.
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MATHEMATICA
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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 *)
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 *)
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PROG
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(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)))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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