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%I #8 Jun 07 2023 11:02:41
%S 25,28,36,40,50,68,70,74,94,95,98,116,119,134,142,143,154,162,170,175,
%T 182,189,190,200,220,226,242,245,262,273
%N Integers whose unitary aliquot sequences are longer than their ordinary aliquot sequences.
%C Here the length of an aliquot sequence is defined to be the length of the transient part of its trajectory + the length of its terminal cycle.
%D Riele, H. J. J. te; Unitary Aliquot Sequences. MR 139/72, Mathematisch Centrum, 1972, Amsterdam.
%D Riele, H. J. J. te; Further Results On Unitary Aliquot Sequences. NW 2/73, Mathematisch Centrum, 1973, Amsterdam.
%H Manuel Benito and Juan L. Varona, <a href="https://doi.org/10.1090/S0025-5718-99-00991-6">Advances In Aliquot Sequences</a>, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
%H Wolfgang Creyaufmueller, <a href="http://www.aliquot.de/aliquote.htm">Aliquot Sequences</a>.
%F Sequence gives those values of n for which A097032(n)>A098007(n).
%e a(5)=50 because the fifth integer whose unitary aliquot sequence is longer than its ordinary aliquot sequence is 50.
%t UnitaryDivisors[n_Integer?Positive]:=Select[Divisors[n],GCD[ #,n/# ]==1&];sstar[n_]:=Plus@@UnitaryDivisors[n]-n;g[n_] := If[n > 0, sstar[n], 0];UnitaryTrajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]];s[n_]:=DivisorSigma[1,n]-n;h[n_] := If[n > 0, s[n], 0];OrdinaryTrajectory[n_] := Most[NestWhileList[h, n, UnsameQ, All]];Select[Range[275],Length[UnitaryTrajectory[ # ]]>Length[OrdinaryTrajectory[ # ]] &]
%Y Cf. A127161, A127162, A127163, A127164, A063769, A063990, A097032, A098007, A097010, A127653, A098185, A127654, A063991, A127655, A097037, A097036.
%K hard,nonn
%O 1,1
%A _Ant King_, Jan 24 2007
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