A096335 Number of iterations of n -> n + tau(n) needed for the trajectory of n to join the trajectory of A064491, or -1 if the two trajectories never merge.

0, 0, 2, 0, 1, 3, 0, 1, 0, 2, 8, 0, 7, 1, 6, 5, 6, 0, 5, 3, 4, 3, 4, 0, 3, 2, 13, 2, 13, 1, 12, 0, 11, 1, 10, 8, 10, 0, 9, 7, 9, 0, 8, 1, 7, 1, 8, 6, 7, 0, 6, 6, 6, 5, 5, 0, 4, 5, 4, 26, 3, 4, 2, 0, 2, 3, 2, 3, 1, 2, 0, 25, 0, 2, 0, 2, 1, 1, 1, 1, 0, 1, 39, 24, 38

Offset: 1

Author

Jason Earls, Jun 28 2004

Keywords

• nonn

Comments

Conjecture: For any positive integer starting value n, iterations of n -> n + tau(n) will eventually join A064491 (verified for all n up to 50000).

The graph looks like a forest of stalks. The tops of the stalks form A036434. - N. J. A. Sloane, Jan 17 2013

Examples

			a(6)=3 because the trajectory for 1 (sequence A064491) starts
1->2->4->7->9->12->18->24->32->38->42...
and the trajectory for 6 starts
6->10->14->18->24->32->38->42->50->56...
so the sequence beginning with 6 joins A064491 after 3 steps.
		

References

• Claudia Spiro, Problem proposed at West Coast Number Theory Meeting, 1977. - From N. J. A. Sloane, Jan 11 2013

Links

• T. D. Noe, Table of n, a(n) for n = 1..11000
• T. D. Noe, Logarithmic plot of 10^6 terms

Crossrefs

Cf. A000005, A036434, A064491, A096335.

Programs

Mathematica

s = 1; t = Join[{s}, Table[s = s + DivisorSigma[0, s], {n, 2, 1000}]]; mx = Max[t]; Table[r = n; gen = 0; While[r < mx && ! MemberQ[t, r], gen++; r = r + DivisorSigma[0, r]]; If[r >= mx, gen = -1]; gen, {n, 100}] (* T. D. Noe, Jan 13 2013 *)

Extensions

Escape clause added to definition by N. J. A. Sloane, Nov 09 2020