A127164 Integers whose aliquot sequences terminate by encountering the prime 7. Also known as the prime family 7.

7, 8, 10, 14, 20, 22, 34, 38, 49, 62, 75, 118, 148, 152, 169, 188, 213, 215

Offset: 1

Author

Ant King, Jan 07 2007

Keywords

• hard
• nonn

Comments

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.

Examples

			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
		

References

• 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.

Links

• Manuel Benito and Juan L. Varona, Advances In Aliquot Sequences, Mathematics of Computation, Vol. 68, No. 225, (1999), pp. 389-393.
• Wolfgang Creyaufmueller, Aliquot sequences.

Crossrefs

Cf. A080907, A127161, A127162, A127163, A098007, A121507, A098008, A007906, A063769, A115060, A115350.

Programs

Mathematica

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] &]

Formula

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.