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A129657
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Infinitary deficient numbers: integers for which A126168(n) < n, or equivalently for which A049417(n) < 2n.
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8
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1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 84
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OFFSET
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1,2
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COMMENTS
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For large n, the distribution of a(n) is approximately linear and asymptotically satisfies a(n)~1.144n. It follows that the density of the infinitary deficient numbers is 1/1.144, which is about 0.874.
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LINKS
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EXAMPLE
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The sixth integer that exceeds its proper infinitary divisor sum is 7. Hence a(6)=7.
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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; InfinitaryDeficientNumberQ[k_]:=If[properinfinitarydivisorsum[k]<k, True, False]; Select[Range[100], InfinitaryDeficientNumberQ[ # ] &] (* 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[100], isigma[#] < 2 # &] (* Amiram Eldar, Jun 09 2019 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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