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A127663
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Infinitary aspiring numbers.
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3
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30, 42, 54, 66, 72, 78, 100, 140, 148, 152, 192, 194, 196, 208, 220, 238, 244, 252, 268, 274, 292, 296, 298, 300, 336, 348, 350, 360, 364, 372, 374, 380, 382, 386, 400, 416, 420, 424, 476, 482, 492, 516, 520, 532, 540, 542, 544, 550, 572, 576, 578, 586, 592
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OFFSET
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1,1
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COMMENTS
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Numbers whose infinitary aliquot sequences end in an infinitary perfect number, but are not infinitary perfect numbers themselves.
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LINKS
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EXAMPLE
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a(5) = 72 because the fifth non-infinitary perfect number whose infinitary aliquot sequence ends in an infinitary perfect number is 72.
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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; g[n_] := If[n > 0, properinfinitarydivisorsum[n], 0]; iTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; InfinitaryPerfectNumberQ[0]=False; InfinitaryPerfectNumberQ[k_Integer] :=If[properinfinitarydivisorsum[k]==k, True, False]; Select[Range[750], InfinitaryPerfectNumberQ[Last[iTrajectory[ # ]]] && !InfinitaryPerfectNumberQ[ # ]&]
f[p_, e_] := Module[{b = IntegerDigits[e, 2]}, m = Length[b]; Product[If[b[[j]] > 0, 1 + p^(2^(m - j)), 1], {j, 1, m}]]; s[n_] := Times @@ f @@@ FactorInteger[n] - n; s[0] = s[1] = 0; q[n_] := Module[{v = NestWhileList[s, n, UnsameQ, All]}, n != v[[-2]] == v[[-1]] > 0]; Select[Range[839], q] (* Amiram Eldar, Mar 11 2023 *)
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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STATUS
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approved
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