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%I #9 Jun 07 2023 10:30:18
%S 7,8,10,14,20,22,34,38,49,62,75,118,148,152,169,188,213,215
%N Integers whose aliquot sequences terminate by encountering the prime 7. Also known as the prime family 7.
%C This sequence is complete only as far as the last term given, for the eventual fate of the aliquot sequence generated by 276 is not (yet) known.
%D Benito, Manuel; Creyaufmueller, Wolfgang; Varona, Juan Luis; and Zimmermann, Paul; Aliquot Sequence 3630 Ends After Reaching 100 Digits; Experimental Mathematics, Vol. 11, No. 2, Natick, MA, 2002, pp. 201-206.
%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 Define s(i)=sigma(i)-i=A000203(i)-i. Then if the aliquot sequence obtained by repeatedly applying the mapping i->s(i) terminates by encountering the prime 7 as a member of its trajectory, i is included in this sequence.
%e a(5)=20 because the fifth integer whose aliquot sequence terminates by encountering the prime 7 as a member of its trajectory is 20. The complete aliquot sequence generated by iterating the proper divisors of 15 is 20->22->14->10->8->7->1->0
%t s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[275], MemberQ[Trajectory[ # ], 7] &]
%Y Cf. A080907, A127161, A127162, A127163, A098007, A121507, A098008, A007906, A063769, A115060, A115350.
%K hard,nonn
%O 1,1
%A _Ant King_, Jan 07 2007
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