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

1,3

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

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

Example

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.

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 *)

Crossrefs

Cf. A000005, A036434, A064491, A096335.

Sequence in context: A328376 A141097 A278045 * A191910 A129503 A225682

Adjacent sequences: A096332 A096333 A096334 * A096336 A096337 A096338

Keyword

nonn

Author

Jason Earls, Jun 28 2004

Extensions

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

Status

approved