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A127162 Composite numbers whose aliquot sequences terminate by encountering a prime number.

%I A127162 #8 Jun 07 2023 11:02:49
%S A127162 4,8,9,10,12,14,15,16,18,20,21,22,24,26,27,30,32,33,34,35,36,38,39,40,
%T A127162 42,44,45,46,48,49,50,51,52,54,55,56,57,58,60,62,63,64,65,66,68,69,70,
%U A127162 72,74,75,76,77,78,80,81,82,84,85,86,87,88,90,91,92,93,94,96,98,99
%N A127162 Composite numbers whose aliquot sequences terminate by encountering a prime number.
%D A127162 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 A127162 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 A127162 Wolfgang Creyaufmueller, <a href="http://www.aliquot.de/aliquote.htm">Aliquot sequences</a>.
%F A127162 Define s(i)=sigma(i)-i=A000203(i)-i. Then if i is composite and the aliquot sequence obtained by repeatedly applying the mapping i->s(i) contains a prime as a member of its trajectory, i is included in this sequence.
%e A127162 a(5)=12 because the fifth composite number whose aliquot sequence terminates by encountering a prime as a member of its trajectory is 12. The complete aliquot sequence generated by iterating the proper divisors of 12 is 12->16->15->9->4->3->1->0
%t A127162 s[n_] := DivisorSigma[1, n] - n; g[n_] := If[n > 0, s[n], 0]; Trajectory[n_] := Most[NestWhileList[g, n, UnsameQ, All]]; Select[Range[2, 275], ! PrimeQ[ # ] && Last[Trajectory[ # ]] == 0 &]
%Y A127162 Cf. A080907, A127161, A127163, A127164, A098007, A121507, A098008, A007906, A063769, A115060, A115350.
%K A127162 nonn
%O A127162 1,1
%A A127162 _Ant King_, Jan 06 2007